Consider the case of a smooth O_(K)\mathcal{O}_{K}-scheme XX as before. Bhatt-Morrow-Scholze suggest in [18, REMARK 1.16] that Z_(p)^(BMS)//p^(n)(r)_(X)\mathbb{Z}_{p}^{\mathrm{BMS}} / p^{n}(r)_{X} should be Schneider's sheaf S_(n)(r)S_{n}(r), and by passage to the limit, there should be a distinguished triangle
{:(6.6)i_(**)WOmega_(log)^(r-1)[-r-1]rarrZ_(p)^(BMS)(r)rarrtau_( <= r)Rj_(**)(Z_(p)(r)_(V,et))rarri_(**)WOmega_(log)^(r-1)[-r]:}\begin{equation*}
i_{*} W \Omega_{\log }^{r-1}[-r-1] \rightarrow \mathbb{Z}_{p}^{\mathrm{BMS}}(r) \rightarrow \tau_{\leq r} R j_{*}\left(\mathbb{Z}_{p}(r)_{V, \mathrm{et}}\right) \rightarrow i_{*} W \Omega_{\log }^{r-1}[-r] \tag{6.6}
\end{equation*}
This has been proven in a work-in-progress by Bhargav Bhatt and Akhil Mathew [16]. They construct an isomorphism of a version of Z_(p)^(BMS)//p^(n)(r)_(X)\mathbb{Z}_{p}^{\mathrm{BMS}} / p^{n}(r)_{X} with Sato's sheaf T_(n)(r)_(X)\mathfrak{T}_{n}(r)_{X} in the semi-stable case; using Zhong's extension of Geisser's results, this gives an isomorphism
I am not aware of a categorical framework for the tower Fil^(n)TC(( tilde(A));Z_(p))\operatorname{Fil}^{n} \mathrm{TC}\left(\tilde{A} ; \mathbb{Z}_{p}\right) and its layers Z_(p)^(BMS)(r)\mathbb{Z}_{p}^{\mathrm{BMS}}(r), analogous to the framework for Voevodsky's slice tower for KK-theory given by SH(k)\mathrm{SH}(k). As A^(1)\mathbb{A}^{1}-homotopy invariance fails for these theories, one would need a stable homotopy theory with a weaker invariance property, perhaps modeled on the one of the categories of motives with modulus discussed in the previous section, for these theories to find a home, in which the Bhatt-Morrow-Scholze tower (6.5) would be seen as a parallel to Voevodsky's slice tower.
FUNDING
The author gratefully acknowledges support from the DFG through the SPP 1786, and through a project that has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 832833).
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In 1999, Khovanov showed that a link invariant known as the Jones polynomial is the Euler characteristic of a homology theory. The knot categorification problem is to find a general construction of knot homology groups, and to explain their meaning - what are they homologies of?
Homological mirror symmetry, formulated by Kontsevich in 1994, naturally produces hosts of homological invariants. Typically though, it leads to invariants which have no particular interest outside of the problem at hand.
I showed recently that there is a new family of mirror pairs of manifolds, for which homological mirror symmetry does lead to interesting invariants and solves the knot categorification problem. The resulting invariants are computable explicitly for any simple Lie algebra, and certain Lie superalgebras.
There are many beautiful strands that connect mathematics and physics. Two of the most fruitful ones are knot theory and mirror symmetry. I will describe a new connection between them. We will find a solution to the knot categorification problem as a new application of homological mirror symmetry.
1.1. Quantum link invariants
In 1984, Jones constructed a polynomial invariant of links in R^(3)\mathbb{R}^{3} [42]. The Jones polynomial is defined by picking a projection of the link to a plane, the skein relation it satisfies
where n=2n=2, and the value for the unknot. It has the same flavor as the Alexander polynomial, dating back to 1928 [8], which one gets by setting n=0n=0 instead.
The proper framework for these invariants was provided by Witten in 1988, who showed that they originate from three-dimensional Chern-Simons theory based on a Lie algebra ^(L)g{ }^{L} \mathrm{~g} [82]. In particular, the Jones polynomial comes from ^(L)g=su_(2){ }^{L} \mathrm{~g}=\mathfrak{s u}_{2} with links colored by the defining two-dimensional representation. The Alexander polynomial comes from the same setting by taking ^(L)g{ }^{L} \mathrm{~g} to be a Lie superalgebra gll_(1∣1)\mathfrak{g l} \mathfrak{l}_{1 \mid 1}. The resulting link invariants are known as the U_(q)(^(L)(g))U_{\mathrm{q}}\left({ }^{L} \mathrm{~g}\right) quantum group invariants. The relation to quantum groups was discovered by Reshetikhin and Turaev [67].
1.2. The knot categorification problem
The quantum invariants of links are Laurant polynomials in q^(1//2)q^{1 / 2}, with integer coefficients. In 1999, Khovanov showed [48, 49] that one can associate to a projection of the link to a plane a bigraded complex of vector spaces
are independent of the choice of projection; they are themselves link invariants.
1.2.1.
Khovanov's construction is part of the categorification program initiated by Crane and Frenkel [25], which aims to lift integers to vector spaces and vector spaces to categories.
A toy model of categorification comes from a Riemannian manifold MM, whose Euler characteristic
is categorified by the cohomology H^(k)(M)=ker d_(k)//im d_(k-1)\mathscr{H}^{k}(M)=\operatorname{ker} d_{k} / \operatorname{im} d_{k-1} of the de Rham complex
The Euler characteristic is, from the physics perspective, the partition function of supersymmetric quantum mechanics with MM as a target space chi(M)=Tr(-1)^(F)e^(-beta H)\chi(M)=\operatorname{Tr}(-1)^{F} e^{-\beta H}, with Laplacian H=dd^(**)+d^(**)dH=d d^{*}+d^{*} d as the Hamiltonian, and d=sum_(k)d_(k)d=\sum_{k} d_{k} as the supersymmetry operator. If hh is a Morse function on MM, the complex can be replaced by a Morse-Smale-Witten complex C_(h)^(**)C_{h}^{*} with the differential d_(h)=e^(h)de^(-h)d_{h}=e^{h} d e^{-h}. The complex C_(h)^(**)C_{h}^{*} is the space of perturbative ground states of a sigma\sigma-model on MM with potential hh [81]. The action of the differential d_(h)d_{h} is generated by solutions to flow equations, called instantons.
1.2.2.
Khovanov's remarkable categorification of the Jones polynomial is explicit and easily computable. It has generalizations of similar flavor for ^(L)g=u_(n){ }^{L} \mathfrak{g}=\mathfrak{} u_{n}, and links colored by its minuscule representations [51].
In 2013, Webster showed [78] that for any ^(L)g{ }^{L} \mathrm{~g}, there exists an algebraic framework for categorification of U_(q)(^(L)(g))U_{\mathrm{q}}\left({ }^{L} \mathrm{~g}\right) invariants of links in R^(3)\mathbb{R}^{3}, based on a derived category of modules of an associative algebra. The KLRW algebra, defined in [78], generalizes the algebras of Khovanov and Lauda [50] and Rouquier [68]. Unlike Khovanov's construction, Webster's categorification is anything but explicit.
1.2.3.
Despite the successes of the program, one is missing a fundamental principle which explains why is categorification possible - the construction has no right to exist. Unlike in our toy example of categorification of the Euler characteristic of a Riemanniann manifold, Khovanov's construction and its generalizations did not come from either geometry or physics in any unified way. The problem Khovanov initiated is to find a general framework for link homology, that works uniformly for all Lie algebras, explains what link homology groups are, and why they exist.
1.3. Homological invariants from mirror symmetry
The solution to the problem comes from a new relation between mirror symmetry and representation theory.
Homological mirror symmetry relates pairs of categories of geometric origin [55]: a derived category of coherent sheaves and a version of the derived Fukaya category, in which complementary aspects of the theory are simple to understand. Occasionally, one can make mirror symmetry manifest, by showing that both categories are equivalent to a derived category of modules of a single algebra.
I will describe a new family of mirror pairs, in which homological mirror symmetry can be made manifest and leads to the solution to the knot categorification problem [1, 2]. Many special features exist in this family, in part due to its deep connections to representation theory. As a result, the theory is solvable explicitly, as opposed to only formally [4,5].
1.4. The solution
We will get not one, but two solutions to the knot categorification problem. The first solution [1] is based on DX\mathscr{D} X, the derived category of equivariant coherent sheaves on a certain holomorphic symplectic manifold X\mathcal{X}, which plays a role in the geometric Langlands correspondence. Recently, Webster proved that D_(X)\mathscr{D}_{X} is equivalent to D_(A)\mathscr{D}_{\mathscr{A}}, the derived category of modules of an algebra A\mathscr{A} which is a cylindrical version of the KLRW algebra from [79,80]. The generalization allows the theory to describe links in R^(2)xxS^(1)\mathbb{R}^{2} \times S^{1}, as well as in R^(3)\mathbb{R}^{3}.
The second solution [2] is based on D_(Y)\mathscr{D}_{Y}, the derived Fukaya-Seidel category of a certain manifold YY with potential WW. The theory generalizes Heegard-Floer theory [63,64[63,64, 66], which categorifies the Alexander polynomial, from ^(L)g=gl_(1∣1){ }^{L} \mathrm{~g}=\mathrm{gl}_{1 \mid 1}, to arbitrary ^(L)g{ }^{L} \mathrm{~g}.
The two solutions are related by equivariant homological mirror symmetry, which is not an equivalence of categories, but a correspondence of objects and morphisms coming from a pair of adjoint functors. In DX\mathscr{D} X, we will learn which question we need to ask to obtain U_(q)(^(L)(g))U_{\mathrm{q}}\left({ }^{L} \mathrm{~g}\right) link homology. In D_(Y)\mathscr{D}_{Y}, we will learn how to answer it.
In [5], we give an explicit algorithm for computing homological link invariants from D_(Y)\mathscr{D}_{Y}, for any simple Lie algebra ^(L)g{ }^{L} \mathrm{~g} and links colored by its minuscule representations. It has an extension to Lie superalgebras ^(L)g=gl_(m∣n){ }^{L} \mathfrak{g}=\mathfrak{g} \mathfrak{l}_{m \mid n} and sp_(m∣2n)\mathfrak{s} \mathfrak{p}_{m \mid 2 n}. In [4], we set the mathematical foundations of D_(Y)\mathscr{D}_{Y} and prove (equivariant) homological mirror symmetry relating it to DX\mathscr{D} X.
2. KNOT INVARIANTS AND CONFORMAL FIELD THEORY
Most approaches to categorification of U_(q)(^(L)(g))U_{\mathrm{q}}\left({ }^{L} \mathrm{~g}\right) link invariants start with quantum groups and their modules. We will start by recalling how quantum groups came into the story. The seeming detour will help us understand how U_(q)(^(L)g)U_{\mathfrak{q}}\left({ }^{L} \mathfrak{g}\right) link invariants arise from geometry, and what categorifies them.
2.1. Knizhnik-Zamolodchikov equation and quantum groups
Chern-Simons theory associates to a punctured Riemann surface A\mathscr{A} a vector space, its Hilbert space. As Witten showed [82], the Hilbert space is finite dimensional, and spanned by vectors that have a name. They are known as conformal blocks of the affine Lie algebra widehat(L)_(g_(kappa))\widehat{L}_{\mathrm{g}_{\kappa}}. The effective level kappa\kappa is an arbitrary complex number, related to qq by q=e^((2pi i)/(kappa))q=e^{\frac{2 \pi i}{\kappa}}. While in principle arbitrary representations of ^(L)g{ }^{L} \mathrm{~g} can occur, in relating to geometry and categorification, we will take them to be minuscule.
To get invariants of knots in R^(3)\mathbb{R}^{3}, one typically takes A\mathscr{A} to be a complex plane with punctures. It is equivalent, but for our purposes better, to take A\mathscr{A} to be an infinite complex cylinder. This way, we will be able to describe invariants of links in R^(2)xxS^(1)\mathbb{R}^{2} \times S^{1}, as well.
2.1.1.
Every conformal block, and hence every state in the Hilbert space, can be obtained explicitly as a solution to a linear differential equation discovered by Knizhnik and Zamolodchikov in 1984 [53]. The KZ equation we get is of trigonometric type, schematically
since A\mathscr{A} is an infinite cylinder. Here, del_(i)=a_(i)(del)/(dela_(i))\partial_{i}=a_{i} \frac{\partial}{\partial a_{i}}, where a_(i)a_{i} is any of the nn punctures in the interior of A\mathcal{A}, colored by a representation V_(i)V_{i} of ^(L)g{ }^{L} \mathrm{~g}. The right hand side of (2.1) is given in terms of classical rr-matrices of ^(L)g{ }^{L} \mathrm{~g}, and acts irreducibly on a subspace of V_(1)ox cdots oxV_(n)V_{1} \otimes \cdots \otimes V_{n} of a fixed weight nu\nu, where V\mathcal{V} takes values [32,33].
The KZ\mathrm{KZ} equations define a flat connection on a vector bundle over the configuration space of distinct points a_(1),dots,a_(n)inAa_{1}, \ldots, a_{n} \in \mathcal{A}. The flatness of the connection is the integrability condition for the equation.
2.1.2.
The monodromy problem of the KZ\mathrm{KZ} equation, which is to describe analytic continuation of its fundamental solution along a path in the configuration space, has an explicit solution. Drinfeld [30] and Kazhdan and Lustig [47] proved that that the monodromy matrix B\mathfrak{B} of the KZ\mathrm{KZ} connection is a product of RR-matrices of the U_(q)(^(L)(g))U_{\mathrm{q}}\left({ }^{L} \mathrm{~g}\right) quantum group corresponding to ^(L)g{ }^{L} \mathrm{~g}. The RR-matrices describe reorderings of neighboring pairs of punctures.
The monodromy matrix B\mathfrak{B} is the Chern-Simons path integral on Axx[0,1]\mathscr{A} \times[0,1] in presence of a colored braid. By composing braids, we get a representation of the affine braid group based on the U_(q)(^(L)(g))U_{\mathrm{q}}\left({ }^{L} \mathrm{~g}\right) quantum group, acting on the space of solutions to the KZ\mathrm{KZ} equation. The braid group is affine, since A\mathscr{A} is a cylinder and not a plane.
2.1.3.
Any link can be represented as a plat closure of some braid. The Chern-Simons path integral together with the link computes a very specific matrix element of the braiding matrix B\mathfrak{B}, picked out by a pair of states in the Hilbert space corresponding to the top and the bottom of Figure 1 .
FIGURE 1
Every link arises as a plat closure of a braid.
These states, describing a collection of cups and caps, are very special solutions of the KZ\mathrm{KZ} equation in which pairs of punctures, colored by conjugate representations V_(i)V_{i} and V_(i)^(**)V_{i}^{*}, come together and fuse to disappear. In this way, both fusion and braiding enter the problem.
2.2. A categorification wishlist
To categorify U_(q)(^(L)(g))U_{\mathfrak{q}}\left({ }^{L} \mathrm{~g}\right) invariants of links in R^(3)\mathbb{R}^{3}, we would like to associate, to the space of conformal blocks of widehat(L_(G))\widehat{L_{\mathcal{G}}} on the Riemann surface A\mathcal{A}, a bigraded category, which in addition to the cohomological grading has a grading associated to q\mathrm{q}. Additional rk(^(L)(g))\mathrm{rk}\left({ }^{L} \mathrm{~g}\right) gradings are needed to categorify invariants of links in R^(2)xxS^(1)\mathbb{R}^{2} \times S^{1}, as they depend on the choice of a flat ^(L)g{ }^{L} \mathrm{~g} connection around the S^(1)S^{1}. To braids, we would like to associate functors between the categories corresponding to the top and bottom. To links, we would like to associate a vector space whose elements are morphisms between the objects of the categories associated to the top and bottom, up to the action of the braiding functor. Moreover, we would like to do this in a way that recovers quantum link invariants upon decategorification. One typically proceeds by coming up with a category, and then works to prove that decategorification gives the link invariants one set out to categorify. A virtue of the solutions in [1,2][1,2] is that the second step is automatic.
3. MIRROR SYMMETRY
Mirror symmetry is a string duality which relates sigma\sigma-models on a pair of Calabi-Yau manifolds X\mathcal{X} and y\mathscr{y}. Its mathematical imprint are relations between very different problems in complex geometry of X\mathcal{X} ("B-type") and symplectic geometry of yy ("A-type"), and vice versa.
Mirror symmetry was discovered as a duality of sigma\sigma-models on closed Riemann surfaces DD. In string theory, one must allow Riemann surfaces with boundaries. This enriches the theory by introducing "branes," which are boundary conditions at del D\partial D and naturally objects of a category [9].
By asking how mirror symmetry acts on branes turned out to yield deep insights into mirror symmetry. One such insight is due to Strominger, Yau, and Zaslov [75], who showed that in order for every point-like brane on X\mathcal{X} to have a mirror on YY, mirror manifolds have to be fibered by a pair of (special Lagrangian) dual tori TT and T^(vv)T^{\vee}, over a common base.
3.1. Homological mirror symmetry
Kontsevich conjectured in his 1994 ICM address [55] that mirror symmetry should be understood as an equivalence of a pair of categories of branes, one associated to complex geometry of X\mathcal{X}, the other to symplectic geometry of y\mathscr{y}.
The category of branes associated to complex geometry of X\mathcal{X} is the derived category of coherent sheaves,
Its objects are "B-type branes," supported on complex submanifolds of X\mathcal{X}. The category of branes associated to symplectic geometry is the derived Fukaya category
Dy=D Fuk(y)\mathscr{D} y=D \operatorname{Fuk}(y)
whose objects are "A-type branes," supported on Lagrangian submanifolds of yy, together with a choice of orientation and a flat bundle. For example, mirror symmetry should map the structure sheaf of a point in X\mathcal{X} to a Lagrangian brane in yy supported on a T^(vv)T^{\vee} fiber. The choice of a flat U(1)U(1) connection is the position of the point in the dual fiber TT.
Kontsevich' homological mirror symmetry is a conjecture that the category of Bbranes on X\mathcal{X} and the category of A-branes on yy are equivalent,
Dx~=Dy\mathscr{D} x \cong \mathscr{D} y
and that this equivalence characterizes what mirror symmetry is.
3.2. Quantum differential equation and its monodromy
Knizhnik-Zamolodchikov equation, which plays a central role in knot theory, has a geometric counterpart. In the world of mirror symmetry, there is an equally fundamental differential equation,
The equation is known as the quantum differential equation of X\mathcal{X}. Both the equation and its monodromy problem featured prominently, starting with the first papers on mirror symmetry, see [37] for an early account. In (3.1), (C_(i))_(alpha)^(beta)=C_(gamma_(i)alpha delta)eta^(delta beta)\left(C_{i}\right)_{\alpha}^{\beta}=C_{\gamma_{i} \alpha \delta} \eta^{\delta \beta} is a connection on a vector bundle with fibers H^("even ")(X)=bigoplus_(k)H^(k)(X,^^^(k)T_(X)^(**))H^{\text {even }}(\mathcal{X})=\bigoplus_{k} H^{k}\left(\mathcal{X}, \wedge^{k} T_{X}^{*}\right) over the complexified Kahler moduli space. The derivative stands for del_(i)=a_(i)(del)/(dela_(i))\partial_{i}=a_{i} \frac{\partial}{\partial a_{i}}, so that del_(i)a^(d)=(gamma_(i),d)a^(d)\partial_{i} a^{d}=\left(\gamma_{i}, d\right) a^{d} for a curve of degree d inH_(2)(X)d \in H_{2}(\mathcal{X}). The connection comes from quantum multiplication with classes gamma_(i)inH^(2)(X)\gamma_{i} \in H^{2}(\mathcal{X}). Given three de Rham cohomology classes on X\mathcal{X}, their quantum product
From the mirror perspective of yy, the connection is the classical Gauss-Manin connection on the vector bundle over the moduli space of complex structures on yy, with fibers the mid-dimensional cohomology H^(d)(y)H^{d}(y) as mirror symmetry identifies H^(k)(X,^^^(k)T_(X)^(**))H^{k}\left(\mathcal{X}, \wedge^{k} T_{X}^{*}\right) with H^(k)(y,^^^(d-k)T_(y)^(**))H^{k}\left(y, \wedge^{d-k} T_{y}^{*}\right).
3.2.1.
Solutions to the equation live in a vector space, spanned by K-theory classes of branes [22,36,41,46]. These are B-type branes on X\mathcal{X}, objects of D_(X)\mathscr{D}_{X}, and A-type branes on YY, objects of Dy\mathscr{D} y. A characteristic feature is that the equation and its solutions mix the A- and B-type structures on the same manifold.
From the perspective of X\mathcal{X}, the solutions of the quantum differential equation come from Gromov-Witten theory. They are obtained by counting holomorphic maps from a domain curve DD to X\mathcal{X}, where DD is best thought of as an infinite cigar [39,40][39,40] together with insertions of a class in alpha inH_("even ")^(**)(X)\alpha \in H_{\text {even }}^{*}(\mathcal{X}) at the origin, and [F]in K(X)[\mathcal{F}] \in K(\mathcal{X}) at infinity. The latter is the K-theory class of a B-type brane FinDx\mathscr{F} \in \mathscr{D} x which serves as the boundary condition at the S^(1)S^{1} boundary at infinity of DD. In the mirror yy, the A- and B-type structures get exchanged. In the interior of DD, supersymmetry is preserved by B-type twist, and at the boundary at infinity we place an A-type brane LinDy\mathscr{L} \in \mathscr{D} y, whose KK-theory class picks which solution of the equation we get.
3.2.2.
One of the key mirror symmetry predictions is that monodromy of the quantum differential equation gets categorified by the action of derived autoequivalences of DX\mathscr{D} X. It is related by mirror symmetry to the monodromy of the Gauss-Manin connection, computed by Picard-Lefshetz theory, whose categorification by Dy\mathscr{D} y is developed by Seidel [71].
The flat section V\mathcal{V} of the connection in (3.1) has a close cousin. This is Douglas' [9,28,29]Pi[9,28,29] \Pi-stability central charge function Z^(0):K(D)rarrC\mathcal{Z}^{0}: K(\mathscr{D}) \rightarrow \mathbb{C}, whose existence motivated Bridgeland's formulation of stability conditions [17]. The Pi\Pi-stability central charge Z^(0)Z^{0} arises from the same setting as V\mathcal{V}, except one places trivial insertions at the origin of DD. This implies that monodromies of V\mathcal{V} and Z^(0)\mathcal{Z}^{0} coincide [22]. In the context of the mirror Y\mathcal{Y}, given any brane LinDy\mathscr{L} \in \mathscr{D} y, its central charge is simply Z^(0)[L]=int_(L)Omega\mathcal{Z}^{0}[\mathscr{L}]=\int_{\mathscr{L}} \Omega, where Omega\Omega is the top holomorphic form on yy. The stable objects are special Lagrangians, on which the phase of Omega\Omega is constant. By mirror symmetry, monodromy of Z^(0)\mathcal{Z}^{0} is expected to induce the action of monodromy on Dx\mathscr{D} x. Examples of braid group actions on the derived categories include works of Khovanov and Seidel [52], Seidel and Thomas [74], and others [18,77].
4. HOMOLOGICAL LINK INVARIANTS FROM B-BRANES
The Knizhnik-Zamolodchikov equation not only has the same flavor as the quantum differential equation, but for some very special choices of X\mathcal{X}, they coincide. For the time being, we will take ^(L)g{ }^{L} \mathrm{~g} to be simply laced, so it coincides with its Langlands dual g\mathrm{g}.
4.1. The geometry
The manifold X\mathcal{X} may be described as the moduli space of GG-monopoles on
with prescribed singularities. The monopole group GG is related to ^(L)G{ }^{L} G, the Chern-Simons gauge group, by Langlands or electric-magnetic duality. In Chern-Simons theory, the knots are labeled by representations of ^(L)G{ }^{L} G and viewed as paths of heavy particles, charged electrically under ^(L)G{ }^{L} G. In the geometric description, the same heavy particles appear as singular, Dirac-type monopoles of the Langlands dual group GG. The fact the magnetic description is what is needed to understand categorification was anticipated by Witten [84-87].
4.1.1.
Place a singular GG monopole for every finite puncture on A~=RxxS^(1)\mathscr{A} \cong \mathbb{R} \times S^{1}, at the point on R\mathbb{R} obtained by forgetting the S^(1)S^{1}. Singular monopole charges are elements of the cocharacter lattice of GG, which Langlands duality identifies with the character lattice of ^(L)G{ }^{L} G. Pick the charge of the monopole to be the highest weight mu_(i)\mu_{i} of the ^(L)G{ }^{L} G representation V_(i)V_{i} coloring the puncture. The relative positions of singular monopoles on R^(3)\mathbb{R}^{3} are the moduli of the metric on X\mathcal{X}, so we will hold them fixed.
The smooth monopole charge is a positive root of ^(L)G{ }^{L} G; choose it so that the total monopole charge is the weight nu\nu of subspace of representation ox_(i)V_(i)\otimes_{i} V_{i}, where the conformal blocks take values. For our current purpose, it suffices to assume
is a dominant weight; ^(L)e_(a){ }^{L} e_{a} are the simple positive roots of ^(L)g{ }^{L} \mathrm{~g}. Provided mu_(i)\mu_{i} are minuscule co-weights of GG and no pairs of singular monopoles coincide, the monopole moduli space X\mathcal{X} is a smooth hyper-Kahler manifold of dimension
It is parameterized, in part, by positions of smooth monopoles on R^(3)\mathbb{R}^{3}.
4.1.2.
A choice of complex structure on X\mathcal{X} reflects a split of R^(3)\mathbb{R}^{3} as RxxC\mathbb{R} \times \mathbb{C}. The relative positions of singular monopoles on C\mathbb{C} become the complex structure moduli, and the relative positions of monopoles on R\mathbb{R} the Kahler moduli.
This identifies the complexified Kahler moduli space of X\mathcal{X} (where the Kahler form gets complexified by a periodic two-form) with the configuration space of nn distinct punctures on A=RxxS^(1)\mathscr{A}=\mathbb{R} \times S^{1}, modulo overall translations, as in Figure 2 .
FIGURE 2
Punctures on A\mathcal{A} correspond to singular GG-monopoles on RinRxxC\mathbb{R} \in \mathbb{R} \times \mathbb{C}.
4.1.3.
As a hyper-Kahler manifold, X\mathcal{X} has more symmetries than a typical Calabi-Yau. For its quantum cohomology to be nontrivial, and for the quantum differential equation to coincide with the KZ\mathrm{KZ} equation, we need to work equivariantly with respect to a torus action that scales its holomorphic symplectic form
where Lambda\Lambda preserves the holomorphic symplectic form, and comes from the Cartan torus of GG. The equivariant parameters of the Lambda\Lambda-action correspond to the choice of a flat ^(L)G{ }^{L} G connection of Chern-Simons theory on R^(2)xxS^(1)\mathbb{R}^{2} \times S^{1}.
4.1.4.
The same manifold X\mathcal{X} has appeared in mathematics before, as a resolution of a transversal slice in the affine Grassmannian Gr_(G)=G((z))//G[[z]]\operatorname{Gr}_{G}=G((z)) / G[[z]] of GG, often denoted by
The two are related by thinking of monopole moduli space X\mathcal{X} as obtained by a sequence of Hecke modifications of holomorphic GG-bundles on C\mathbb{C} [45].
Manifold X\mathcal{X} is also the Coulomb branch of a 3d quiver gauge theory with N=4\mathcal{N}=4 supersymmetry, with quiver based on the Dynkin diagram of gg, see e.g. [19]. The ranks of the flavor and gauge symmetry groups are determined from the weights mu\mu and nu\nu.
4.1.5.
The vector vec(mu)=(mu_(1),dots,mu_(n))\vec{\mu}=\left(\mu_{1}, \ldots, \mu_{n}\right) in (4.3) encodes singular monopole charges, and the order in which they appear on R\mathbb{R}, and nu\nu is the total monopole charge. The ordering of entries of vec(mu)\vec{\mu} is a choice of a chamber in the Kahler moduli. We will suppress vec(mu)\vec{\mu} for the most part, and denote all the corresponding distinct symplectic manifolds by X\mathcal{X}.
4.1.6.
By a recent theorem of Danilenko [26], the Knizhnik-Zamolodchikov equation corresponding to the Riemann surface A=RxxS^(1)\mathcal{A}=\mathbb{R} \times S^{1}, with punctures colored by minuscule representations V_(i)V_{i} of ^(L)g{ }^{L} \mathrm{~g}, coincides with the quantum differential equation of the TT-equivariant Gromov-Witten theory on X=Gr^( vec(mu))_(nu)\mathcal{X}=\mathrm{Gr}^{\vec{\mu}}{ }_{\nu}.
This has many deep consequences.
4.2. Branes and braiding
Since the KZ\mathrm{KZ} equation is the quantum-differential equation of TT-equivariant Gromov-Witten theory of X\mathcal{X}, the space of its solutions gets identified with K_(T)(X)K_{T}(\mathcal{X}), the TT-equivariant KK-theory of X\mathcal{X}.
This is the KK-group of the category of its B-type branes, the derived category of TT-equivariant coherent sheaves on X\mathcal{X},
This connection between the KZ\mathrm{KZ} equation and Dx\mathscr{D} x is the starting point for our first solution of the categorification problem.
4.2.1.
A colored braid with nn strands in Axx[0,1]\mathcal{A} \times[0,1] has a geometric interpretation as a path in the complexified Kahler moduli of X\mathcal{X} that avoids singularities, as the order of punctures on A\mathcal{A} corresponds to a choice of chamber in the Kahler moduli of X\mathcal{X}.
The monodromy of the quantum differential equation along this path acts on K_(T)(X)K_{T}(\mathcal{X}). Since the quantum differential equation coincides with the KZ\mathrm{KZ} equation, by the theorem of [26], K_(T)(X)K_{T}(\mathcal{X}) becomes a module for U_(q)(^(L)(g))U_{\mathrm{q}}\left({ }^{L} \mathrm{~g}\right), corresponding to the weight nu\nu subspace of representation V_(1)ox cdots oxV_(n)V_{1} \otimes \cdots \otimes V_{n}
The fact that derived equivalences of Dx\mathscr{D} x categorify this action is not only an expectation, but also a theorem by Bezrukavnikov and Okounkov [14], whose proof makes use of quantization of X\mathcal{X} in characteristic pp.
4.2.2.
From physics perspective, the reason derived equivalences of Dx\mathscr{D} x had to categorify the action of monodromy of the quantum differential equation on K_(T)(X)K_{T}(\mathcal{X}) is as follows.
Braid group acts, in the sigma\sigma-model on the cigar DD from Section 3.2.1, by letting the moduli of X\mathcal{X} vary according to the braid near the boundary at infinity. The Euclidean time, running along the cigar, is identified with the time along the braid. This leads to a Berry phase-type problem studied by Cecotti and Vafa [22]. It follows that the sigma\sigma-model on the annulus, with moduli that vary according to the braid, computes the matrix element of the monodromy B\mathfrak{B}, picked out by a pair of branes F_(0)\mathscr{F}_{0} and F_(1)\mathscr{F}_{1} at its boundaries.
The sigma\sigma-model on the same Euclidian annulus, where we take the time to run around S^(1)S^{1} instead, computes the index of the supercharge QQ preserved by the two branes. The cohomology of QQ is computed by Dx\mathscr{D} x as its most basic ingredient, the space of morphisms
between the branes. This is the space of supersymmetric ground states of the sigma\sigma-model on a strip, obtained by cutting the annulus open. We took here all the variations of moduli to happen near one boundary, at the expense of changing a boundary condition from F_(0)\mathscr{F}_{0} to BF_(0)\mathscr{B} \mathcal{F}_{0}. This does not affect the homology [1,35][1,35], for the very same reason the theory depends on the homotopy type of the braid only. Per construction, the graded Euler characteristic of the homology theory, computed by closing the strip back up to the annulus, is the braiding matrix element,
manifestly categorifies the monodromy matrix BinU_(q)(^(L)(g))\mathfrak{B} \in U_{\mathfrak{q}}\left({ }^{L} \mathrm{~g}\right) of the KZ\mathrm{KZ} equation.
4.3. Link invariants from perverse equivalences
The quantum U_(q)(^(L)(g))U_{\mathcal{q}}\left({ }^{L} \mathrm{~g}\right) invariants of knots and links are matrix elements of the braiding matrix B\mathscr{B}, so they too will be categorified by Dx\mathscr{D} x, provided we can identify objects U inDxU \in \mathscr{D} x which serve as cups and caps.
Conformal blocks corresponding to cups and caps are defined using fusion [62]. The geometric analogue of fusion, in terms of X\mathcal{X} and its category of branes, was shown in [1] to be the existence of certain perverse filtrations on Dx\mathscr{D} x, defined by abstractly by Chuang and Rouqiuer [24]. The utility of perverse filtrations for understanding the action of braiding on Dx\mathscr{D} x parallels the utility of fusion in describing the action of braiding in conformal field theory. In particular, it leads to identification of the cup and cap branes U\boldsymbol{U} we need, and a
4.3.1.
As we bring a pair of punctures at a_(i)a_{i} and a_(j)a_{j} on A\mathcal{A} together, we get a new natural basis of solutions to the KZ\mathrm{KZ} equation, called the fusion basis, whose virtue is that it diagonalizes braiding. The possible eigenvectors are labeled by the representations
that occur in the tensor product of representations V_(i)V_{i} and V_(j)V_{j} labeling the punctures. Because V_(i)V_{i} and V_(j)V_{j} are minuscule representations, the nonzero multiplicities on the right-hand side are all equal to 1 . The cap arises as a special case, obtained by starting with a pair of conjugate representations V_(i)V_{i} and V_(i)^(***)V_{i}^{\star}, and picking the trivial representation in their tensor product.
4.3.2.
From perspective of X\mathcal{X}, a pair of singular monopoles of charges mu_(i)\mu_{i} and mu_(j)\mu_{j} are coming together on R\mathbb{R}, as in Figure 2, and we approach a wall in Kahler moduli at which X\mathcal{X} develops a singularity. At the singularity, a collection of cycles vanishes. This is due to monopole bubbling phenomena described by Kapustin and Witten in [45].
The types of monopole bubbling that can occur are labeled by representations V_(k_(m))V_{k_{m}} that occur in the tensor product V_(i)oxV_(j)V_{i} \otimes V_{j}. The moduli space of monopoles whose positions we need to tune for the bubbling of type V_(k_(m))V_{k_{m}} to occur is Gr_(mu_(k_(m)))^((mu_(i),mu_(j)))=T^(**)F_(k_(m))\operatorname{Gr}_{\mu_{k_{m}}}^{\left(\mu_{i}, \mu_{j}\right)}=T^{*} F_{k_{m}}, where mu_(k_(m))\mu_{k_{m}} is the highest weight of V_(k_(m))V_{k_{m}}. This space is transverse to the locus where exactly mu_(i)+mu_(j)-mu_(k_(m))\mu_{i}+\mu_{j}-\mu_{k_{m}} monopoles have bubbled off [1]. It has a vanishing cycle F_(k_(m))F_{k_{m}}, corresponding to the representation V_(k_(m))V_{k_{m}}, as its zero section. (Viewing X\mathcal{X} as the Coulomb branch, monopole bubbling is related to partial Higgsing phenomena.)
4.3.3.
Conformal blocks which diagonalize the action of braiding do not in general come from actual objects of the derived category D_(x)\mathscr{D}_{x}. As is well known from Picard-Lefshetz theory, eigensheaves of braiding, branes on which the braiding acts only by degree shifts BE=E[D_(E)]{C_(E)}\mathscr{B} \mathcal{E}=\mathcal{E}\left[D_{\mathcal{E}}\right]\left\{C_{\mathcal{E}}\right\}, are very rare.
by the order of vanishing of the Pi\Pi-stability central charge Z^(0):K(X)rarrC\mathcal{Z}^{0}: K(\mathcal{X}) \rightarrow \mathbb{C}. More precisely, one gets a pair of such filtrations, one on each side of the wall. Crossing the wall preserves the filtration, but has the effect of mixing up branes at a given order in the filtration, with those at lower orders, whose central charge vanishes faster. Because X\mathcal{X} is hyper-Kahler, the Pi\Pi-stability central charge is given in terms of classical geometry (by Eq. (4.7) of [1]).
The existence of the filtration with the stated properties follows from the existence of the equivariant central charge function Z\mathcal{Z},
and the fact the action of braiding on K_(T)(X)K_{T}(\mathcal{X}) lifts to the action on DX\mathscr{D} X, by the theorem of [14]. The equivariant central charge Z\mathcal{Z} is computed by the equivariant Gromov-Witten theory on X\mathcal{X} in a manner analogous to V\mathcal{V}, starting with the sigma\sigma-model on the cigar DD except with no insertion at its tip. It reduces to the Pi\Pi-stability central charge Z^(0)\mathcal{Z}^{0} by turning the equivariant parameters off.
4.3.4.
While B\mathscr{B} has few eigensheaves in Dx\mathscr{D} \mathcal{x}, it acts by degree shifts
on the quotient subcategories. The degree shifts may be read off from the eigenvectors of the action of braiding on the equivariant central charge function Z\mathcal{Z}. As the punctures at a_(i)a_{i} and a_(j)a_{j} come together, the eigenvector corresponding to the representation V_(k_(m))V_{k_{m}} in (4.6), vanishes as [1]
The cohomological degree shift D_(k_(m))=dim_(C)F_(k_(m))D_{k_{m}}=\operatorname{dim}_{\mathbb{C}} F_{k_{m}} is by the dimension of the vanishing cycle. The equivariant degree shift C_(k_(m))C_{k_{m}} is essentially the one familiar from the action of braiding on conformal blocks of widehat(L_(g))\widehat{L_{\mathrm{g}}} in the fusion basis [1].
4.3.5.
The derived equivalences of this type are the perverse equivalences of Chuang and Rouquier [23, 24]. They envisioned them as a way to describe derived equivalences which come from variations of Bridgeland stability conditions, but with few examples from geometry.
Traditionally, braid group actions on derived categories of coherent sheaves, or Bbranes, are fairly difficult to describe, see for example [20,21]. Braid group actions on the categories of A-branes are much easier to understand, via Picard-Lefshetz theory and its categorical uplifts [71], see e.g. [52,77]. The theory of variations of stability conditions, by Douglas and Bridgeland, was invented to bridge the two [9,29][9,29].
4.3.6.
As a by-product, we learn that conformal blocks describing collections cups or caps colored by minuscule representations, come from branes in Dx\mathscr{D} x which have a simple geometric meaning [1][1].
Take X=Gr_(0)^((mu_(1),mu_(1)^(**),dots,mu_(d),mu_(d)^(**)))\mathcal{X}=\operatorname{Gr}_{0}^{\left(\mu_{1}, \mu_{1}^{*}, \ldots, \mu_{d}, \mu_{d}^{*}\right)} corresponding to A\mathcal{A} with n=2dn=2 d punctures, colored by pairs of complex conjugate, minuscule representations V_(i)V_{i} and V_(i)^(**)V_{i}^{*}. We get a vanishing cycle UU in X\mathcal{X} which is a product of dd minuscule Grassmannians,
U=G//P_(1)xx cdots xx G//P_(d)U=G / P_{1} \times \cdots \times G / P_{d}
where P_(i)P_{i} is the maximal parabolic subgroup of GG associated to representation V_(i)V_{i}. This vanishing cycle embeds in X\mathcal{X} as a compact holomorphic Lagrangian, so in the neighborhood of UU, we can model X\mathcal{X} as T^(**)UT^{*} U. The structure sheaf
of UU is the brane we are after. The Grassmannian G//P_(i)G / P_{i} is the cycle that vanishes when a single pair of singular monopoles of charges mu_(i)\mu_{i} and mu_(i)^(**)\mu_{i}^{*} come together, as Gr_(0)^((mu_(i),mu_(i)^(**)))=\operatorname{Gr}_{0}^{\left(\mu_{i}, \mu_{i}^{*}\right)}=T^(**)G//P_(i)T^{*} G / P_{i}.
The brane U\mathcal{U} lives at the very bottom of a dd-fold filtration which Dx\mathscr{D} x develops at the intersection of dd walls in the Kahler moduli of X\mathcal{X} corresponding to bringing punctures together pairwise. It follows U\mathcal{U} is the eigensheaf of braiding each pair of matched endpoints. It is extremely special, for the same reason the trivial representation is special.
4.3.7.
Just as fusion provides the right language to understand the action of braiding in conformal field theory, the perverse filtrations provide the right language to describe the action of braiding on derived categories. Using perverse filtrations and the very special properties of the vanishing cycle branes U inDxU \in \mathscr{D} x, one gets the following theorem [1]:
Theorem 1. For any simply laced Lie algebra ^(L)g{ }^{L} \mathfrak{g}, the homology groups
Hom_(D)^(**,**)(BU,U)\operatorname{Hom}_{\mathscr{D}}^{*, *}(\mathscr{B} U, U)
categorify U_(q)(^(L)g)U_{\mathfrak{q}}\left({ }^{L} \mathfrak{g}\right) quantum link invariants, and are themselves link invariants.
4.3.8.
As an illustration, proving that (the equivalent of) the pitchfork move in the figure below holds in Dx\mathscr{D} x
FIGURE 3
A move equivalent to the pitchfork move.
requires showing that we have a derived equivalence
where C_(i)\mathscr{C}_{i} and C_(i)^('')\mathscr{C}_{i}^{\prime \prime} are cup functors on the right and the left in Figure 3, respectively. They increase the number of strands by two and map
C_(i):D_(X_(n-2))rarrD_(X_(n))quad" and "quadC_(i)^(''):D_(X_(n-2))rarrD_(X_(n)^(''))\mathscr{C}_{i}: \mathscr{D}_{X_{n-2}} \rightarrow \mathscr{D}_{X_{n}} \quad \text { and } \quad \mathscr{C}_{i}^{\prime \prime}: \mathscr{D}_{X_{n-2}} \rightarrow \mathscr{D}_{X_{n}^{\prime \prime}}
where the subscript serves to indicate the number of strands. The functor B\mathscr{B} is the equivalence of categories from the theorem of [14]
corresponding to braiding V_(k)(a_(k))V_{k}\left(a_{k}\right) with V_(i)(a_(i))oxV_(i)^(**)(a_(j))V_{i}\left(a_{i}\right) \otimes V_{i}^{*}\left(a_{j}\right) where V_(i)V_{i} and V_(i)^(**)V_{i}^{*} color the red and V_(k)V_{k} the black strand in Figure 3.
are the subcategories which are the bottom-most part of the double filtrations of Dx_(n)\mathscr{D} x_{n} and D_(X_(n)^(''))\mathscr{D}_{X_{n}^{\prime \prime}}, corresponding to the intersection of walls at which the three punctures come together. By the definition of perverse filtrations, the functor B\mathscr{B} acts at a bottom part of a double filtration at most by degree shifts. The degree shifts are trivial too, since if they were not, the relation we are after would not hold even in conformal field theory, and we know it does. To complete the proof, one recalls that a perverse equivalence that acts by degree shifts that are trivial is an equivalence of categories [24].
Proofs of invariance under the Reidermeister 0 and the framed Reidermeister I moves are similar. The invariance under Reidermeister II and III moves follows from the theorem of [14]. One should compare this to proofs of the same relations in [20, 21], which are more technical and less general.
4.3.9.
An elementary consequence is a geometric explanation of mirror symmetry which relates the U_(q)(^(L)(g))U_{\mathrm{q}}\left({ }^{L} \mathrm{~g}\right) invariants of a link KK and its mirror reflection K^(**)K^{*}.
It is a consequence of a basic property of D_(X)\mathscr{D}_{X}, Serre duality. Serre duality implies the isomorphism of homology groups on X\mathcal{X} which is a 2d2 d complex-dimensional Calabi-Yau manifold,
{:(4.12)Hom_(DX)(BU,U[M]{J_(0),( vec(J))})=Hom_(Dx)(BU,U[2d-M]{-d-J_(0),-( vec(J))}):}\begin{equation*}
\operatorname{Hom}_{\mathscr{D} X}\left(\mathscr{B} U, U[M]\left\{J_{0}, \vec{J}\right\}\right)=\operatorname{Hom}_{\mathscr{D} x}\left(\mathscr{B} U, U[2 d-M]\left\{-d-J_{0},-\vec{J}\right\}\right) \tag{4.12}
\end{equation*}
The equivariant degree shift comes from the fact the unique holomorphic section of the canonical bundle has weight dd under the C_(q)^(xx)subT\mathbb{C}_{\mathrm{q}}^{\times} \subset \mathrm{T} action. Mirror symmetry follows by taking Euler characteristic of both sides [1].
4.4. Algebra from B-branes
Bezrukavnikov and Kaledin, using quantization in characteristic pp, constructed a tilting vector bundle T\mathcal{T}, on any X\mathcal{X} which is a symplectic resolution [12,13,43,44][12,13,43,44]. Its endomorphism algebra
is an ordinary associative algebra, graded only by equivariant degrees. The derived category D_(A)\mathscr{D}_{\mathscr{A}} of its modules is equivalent to D_(X)\mathscr{D}_{X},
Webster recently computed the algebra A\mathscr{A} for our X\mathcal{X} [80], and showed that it coincides with a cylindrical version of the KLRW algebra from [78]. Working with the cylindrical KLRW algebra, as opposed to the ordinary one, leads to invariants of links in R^(2)xxS^(1)\mathbb{R}^{2} \times S^{1} and not just in R^(3)\mathbb{R}^{3}. The KLRW algebra generalizes the algebras of Khovanov and Lauda [50] and Rouquier [68]. The cylindrical version of the KLR algebra corresponds to X\mathcal{X} which is a Coulomb branch of a pure 3D gauge theory.
4.4.1.
The description of link homologies via D_(X)=Coh_(T)(X)\mathscr{D}_{X}=\operatorname{Coh}_{T}(\mathcal{X}) provides a geometric meaning of homological U_(q)(^(L)g)U_{\mathfrak{q}}\left({ }^{L} \mathfrak{g}\right) link invariants. Even so, without further input, the description of link homologies either in terms of D_(X)\mathscr{D}_{X} or D_(A)\mathscr{D}_{\mathscr{A}} is purely formal. With the help of (equivariant) homological mirror symmetry, we will give a description of link homology groups which is explicit and explicitly computable; in this sense, link homology groups come to life in the mirror.
5. MIRROR SYMMETRY FOR MONOPOLE MODULI SPACE
In the very best instances, homological mirror symmetry relating Dy\mathscr{D} y and Dx\mathscr{D} x can be made manifest, by showing that each is equivalent to D_(A)\mathscr{D}_{\mathscr{A}}, the derived category of modules
of the same associative algebra A\mathscr{A},
{:(5.1)D_(x)~=D_(A)~=Dy:}\begin{equation*}
\mathscr{D}_{x} \cong \mathscr{D}_{\mathscr{A}} \cong \mathscr{D} y \tag{5.1}
\end{equation*}
is the endomorphism algebra of a set of branes T=bigoplus_(C)Te\mathcal{T}=\bigoplus_{\mathscr{C}} \mathcal{T} \mathscr{e}, which generate Dx\mathscr{D} x and Dy\mathscr{D} y. For economy, we will be denoting branes related by mirror symmetry by the same letter.
An elementary example [10] is mirror symmetry relating a pair of infinite cylinders, X=C^(xx)\mathcal{X}=\mathbb{C}^{\times}and y=RxxS^(1)y=\mathbb{R} \times S^{1}, whose torus fibers are dual S^(1)S^{1}, s. Both Dx\mathscr{D} x, the derived category of coherent sheaves on X\mathcal{X}, and Dy\mathscr{D} y, based on the wrapped Fukaya category, are generated by a single object T\mathcal{T}, a flat line bundle on X\mathcal{X} and a real-line Lagrangian on Y\mathscr{Y}. Their algebras of open strings are the same, equal to the algebra A=C[x^(+-1)]\mathscr{A}=\mathbb{C}\left[x^{ \pm 1}\right] of holomorphic functions on the cylinder.
X\mathcal{X}
FIGURE 4
A simple example of manifest mirror symmetry.
5.1. The algebra for homological mirror symmetry
In our setting, the generator T\mathcal{T} of Dx\mathscr{D} x is the tilting generator of Bezrukavnikov and Kaledin from Section 4.4. Webster's proof of the equivalence of categorification of U_(q)(^(L)g)U_{\mathfrak{q}}\left({ }^{L} \mathfrak{g}\right) link invariants and B-type branes on X\mathcal{X} and via the cKLRW algebra A\mathscr{A} should be understood as the first of the two equivalences in (5.1).
5.1.1.
The mirror yy of X\mathcal{X} is the moduli space of GG monopoles, of the same charges as XX except on R^(2)xxS^(1)\mathbb{R}^{2} \times S^{1} instead of on R^(3)\mathbb{R}^{3}, with only complex and no Kahler moduli turned on, and equipped with a potential [2]. Without the potential, the mirror to yy would be another moduli space of GG monopoles on R^(2)xxS^(1)\mathbb{R}^{2} \times S^{1}.
The theory based on Dy\mathscr{D} y, the derived Fukaya-Seidel category of yy, is in the same spirit as the work of Seidel and Smith [72]. They pioneered geometric approaches to link homology, but produced a only singly graded theory, known as symplectic Khovanov homology. The computation of Dy\mathscr{D} y, which makes mirror symmetry in (5.1) manifest, is given in the joint work with Danilenko, Li, and Zhou [4].
5.2. The core of the monopole moduli space
Working equivariantly with respect to a C_(q)^(xx)\mathbb{C}_{\mathrm{q}}^{\times}-symmetry which scales the holomorphic symplectic form of X\mathcal{X}, all the information about its geometry should be encoded in a core locus preserved by such actions.
The core XX is a singular holomorphic Lagrangian in X\mathcal{X} which is the union of supports of all stable envelopes [7,61]. Equivalently, XX is the union of all attracting sets of Lambda\Lambda-torus actions on X\mathcal{X}, where we let Lambda\Lambda vary over all chambers. If we view X\mathcal{X} as the monopole moduli space, we can put this more simply: XX is the locus where all the monopoles, singular or not, are at the origin of C\mathbb{C} in RxxC\mathbb{R} \times \mathbb{C}. Viewing it as a Coulomb branch, XX is the locus at which the complex scalar fields in vector multiplets vanish.
We will define the equivariant mirror YY of X\mathcal{X} to be the ordinary mirror of its core, so we have
Working equivariantly with respect to the TT-action on X\mathcal{X}, the equivariant mirror gets a potential WW, making the theory on YY into a Landau-Ginsburg model. While XX embeds into X\mathcal{X} as a holomorphic Lagrangian of dimension d,yd, y fibers over YY with holomorphic Lagrangian (C^(xx))^(d)\left(\mathbb{C}^{\times}\right)^{d} fibers.
5.2.1.
A model example is X\mathcal{X} which is the resolution of an A_(n-1)A_{n-1} hypersurface singularity, uv=z^(n);Xu v=z^{n} ; \mathcal{X} is the moduli space of a single smooth G=SU(2)//Z_(2)G=\mathrm{SU}(2) / \mathbb{Z}_{2} monopole, in the presence of nn singular ones. The core XX is a collection of n-1P^(1)n-1 \mathbb{P}^{1} 's with a pair of infinite discs attached, as in Figure 5.
FIGURE 5
Core XX of a resolution of the A_(n-1)A_{n-1} singularity.
The ordinary mirror yy of X\mathcal{X} is the complex structure deformation of the "multiplicative" A_(n-1)A_{n-1} surface singularity, with a potential which we will not need. yy is a C^(xx)\mathbb{C}^{\times}fibration over YY which is itself an infinite cylinder, a copy of C^(xx)\mathbb{C}^{\times}with nn points deleted. At the marked points, the C^(xx)\mathbb{C}^{\times}fibers degenerate. There are n-1n-1 Lagrangian spheres in yy, which are mirror to n-1P^(1)n-1 \mathbb{P}^{1} 's in XX. They project to Lagrangians in YY which begin and end at the punctures.
5.2.2.
The model example corresponds to ^(L)G=SU(2){ }^{L} G=\mathrm{SU}(2) Chern-Simons theory on R^(2)xxS^(1)\mathbb{R}^{2} \times S^{1}, and widehat(N1_(2))\widehat{\mathfrak{N 1}_{2}} conformal blocks on A=RxxS^(1)\mathcal{A}=\mathbb{R} \times S^{1}. The nn punctures on A\mathcal{A} are colored by the fundamental, two-dimensional representation V_(1//2)V_{1 / 2} of Nu_(2)\mathfrak{N u}_{2}, and we take the subspace of weight
FIGURE 6
Lagrangian spheres in y\boldsymbol{y} mirror the vanishing P^(1)\mathbb{P}^{1} 's in X\mathcal{X}.
one level below the highest. Note that YY coincides with the Riemann surface A\mathcal{A} where the conformal blocks live. This is not an accident.
In the model example, both XX and YY are S^(1)S^{1} fibrations over R\mathbb{R} with nn marked points. At the marked points, the S^(1)S^{1} fibers of XX degenerate. In YY, this is mirrored by fibers that decompactify, due to points which are deleted.
5.2.3.
More generally, for X=Gr^( vec(mu))_(nu)\mathcal{X}=\mathrm{Gr}^{\vec{\mu}}{ }_{\nu} we have d_(a)d_{a} smooth GG-monopoles colored by simple roots ^(L)e_(a){ }^{L} e_{a} and otherwise identical. It follows that the common base of SYZ fibrations of XX and YY is the configuration space of the smooth monopoles on the real line R\mathbb{R} with nn marked points. The marked points are labeled by the weights mu_(i)\mu_{i} of ^(L)g{ }^{L} \mathrm{~g}, which are the singular monopole charges.
An explicit description of YY, as well as its category of A-branes D_(Y)\mathscr{D}_{Y}, is given [4]. Here we will only describe some of its features. In an open set, YY coincides with
the configuration space of d=sum_(a=1)^(rk)d_(a)d=\sum_{a=1}^{r k} d_{a} points on the punctured Riemann surface A\mathcal{A}, "colored" by simple roots ^(L)e_(a){ }^{L} e_{a} of ^(L)g{ }^{L} \mathrm{~g}, but otherwise identical. The open set is the complement of the divisor of zeros and of poles of function f^(0)f^{0} in (5.5).
where y_(alpha,a)y_{\alpha, a} are coordinates on dd copies of A\mathcal{A}, viewed as the complex plane with 0 and oo\infty deleted. While Omega\Omega itself is not globally well defined, so K_(Y)K_{Y} is not trivial, Omega^(ox2)\Omega^{\otimes 2} is well defined and
This allows D_(Y)\mathscr{D}_{Y} to have a Z\mathbb{Z}-valued cohomological grading. The symplectic form on YY is inherited from the symplectic form on yy, by restricting it to the vanishing (S^(1))^(d)\left(S^{1}\right)^{d} in each of its (C^(xx))^(d)\left(\mathbb{C}^{\times}\right)^{d} fibers over YY [4]. The precise choice of symplectic structure is the one compatible
with mirror symmetry which we used to define YY, as the equivariant mirror of X=Gr^( vec(mu))_(nu)\mathcal{X}=\mathrm{Gr}^{\vec{\mu}}{ }_{\nu} and the ordinary mirror of its core.
Including the equivariant TT-equivariant action on X\mathcal{X} and XX corresponds to adding to the sigma\sigma-model on YY a potential
which is a multivalued holomorphic function on Y;lambda_(a)Y ; \lambda_{a} are the equivariant parameters of the Lambda\Lambda-action on X\mathcal{X}, and
q=e^(2pi ilambda_(0))q=e^{2 \pi i \lambda_{0}}
The potentials W^(0)W^{0} and W^(a)W^{a} are given by
The superpotential WW breaks the conformal invariance of the sigma\sigma-model to YY if lambda_(0)!=0\lambda_{0} \neq 0, since only a quasihomogenous superpotential is compatible with it. This is mirror to breaking of conformal invariance on X\mathcal{X} by the C_(q)^(xx)\mathbb{C}_{q}^{\times}-action for q!=1q \neq 1.
Since W^(0)W^{0} and W^(a)W^{a} are multivalued, YY is equipped with a collection of closed oneforms with integer periods
c^(0)=dW^(0)//2pi i,quadc^(a)=dW^(a)//2pi i inH^(1)(Y,Z)c^{0}=d W^{0} / 2 \pi i, \quad c^{a}=d W^{a} / 2 \pi i \in H^{1}(Y, \mathbb{Z})
which introduce additional gradings in the category of A-branes, as in [73].
5.2.4.
From the mirror perspective, the conformal blocks of widehat(L_(g))\widehat{L_{\mathrm{g}}} come from the B-twisted Landau-Ginsburg model (Y,W)(Y, W) on DD which is an infinitely long cigar, with A-type boundary condition at infinity corresponding to a Lagrangian L in YL \in Y. The partition function of the theory has the following form:
where Phi_(alpha)\Phi_{\alpha} are chiral ring operators, inserted at the tip of the cigar [22,39,40]. By placing the trivial insertion at the origin instead, we get the equivariant central charge function Z[L]=\mathcal{Z}[L]=int_(L)Omegae^(-W)\int_{L} \Omega e^{-W}; by further turning the equivariant parameters off, the potential WW vanishes and the equivariant central charge becomes the ordinary brane central charge Z^(0)[L]=int_(L)OmegaZ^{0}[L]=\int_{L} \Omega.
We have (re)discovered, from mirror symmetry, an integral representation of the conformal blocks of widehat(L_(g))\widehat{L_{\mathrm{g}}}. This "free field representation" of conformal blocks, remarkable for its simplicity [32], goes back to the 1980s work of Kohno and Feigin and Frenkel [34,54], and of Schechtman and Varchenko [69,70].
5.2.5.
There is a reconstruction theory, due to Givental [38] and Teleman [76], which says that, starting with the solution of the quantum differential equation or its mirror counterpart, one gets to reconstruct all genus topological string amplitudes for any semisimple 2D field theory. The semisimplicity condition is satisfied in our case, as WW has isolated critical points. It follows the B-twisted Landau-Ginsburg model on (Y,W)(Y, W) and A-twisted TT-equivariant sigma model on X\mathcal{X} are equivalent to all genus [2]. Thus, equivariant mirror symmetry holds as an equivalence of topological string amplitudes.
5.3. Equivariant Fukaya-Seidel category
For every A-brane LL at the boundary at infinity of the cigar DD, we get a solution of the KZ\mathrm{KZ} equation. The brane is an object of
the derived Fukaya-Seidel category of YY with potential WW. The category should be thought of as a category of equivariant A-branes, due to the fact WW in (5.4) is multivalued. Another novel aspect of D_(Y)\mathscr{D}_{Y} is that it provides an example of Fukaya-Seidel category with coefficients in perverse schobers. This structure, inherited from equivariant mirror symmetry, was discovered in [4][4].
5.3.1.
Objects of D_(Y)\mathscr{D}_{Y} are Lagrangians in YY, equipped with some extra data. A Lagrangian in YY is a product of dd one-dimensional curves on A\mathscr{A} which are colored by simple roots and may be immersed; or a simplex obtained from an embedded curve, as a configuration space of dd partially ordered colored points. The theory also includes more abstract branes, which are iterated mapping cones over morphisms between Lagrangians.
5.3.2.
The extra data includes a grading by Maslov and equivariant degrees. The equivariant grading of a brane in D_(Y)\mathscr{D}_{Y} is defined by choosing a lift of the phase of e^(-W)e^{-W} to a real-valued function on the Lagrangian LL. The equivariant degree shift operation,
L rarr L{ vec(d)}L \rightarrow L\{\vec{d}\}
with vec(d)inZ^(rk+1)\vec{d} \in \mathbb{Z}^{r k+1}, corresponds to changing the lift of WW on LL, now viewed as a graded Lagrangian, W|_(L{ vec(d)})=W|_(L)+2pi i vec(lambda)* vec(d)\left.W\right|_{L\{\vec{d}\}}=\left.W\right|_{L}+2 \pi i \vec{\lambda} \cdot \vec{d}. This is analogous to how a choice of a lift of the phase of Omega^(ox2)\Omega^{\otimes 2} defines the Maslov, or cohomological, grading of a Lagrangian. This restricts the Lagrangians that give rise to objects of D_(Y)\mathscr{D}_{Y} to those for which such lifts can be defined.
More generally, branes in D_(Y)\mathscr{D}_{Y} are graded Lagrangians LL equipped with an extra structure of a local system Lambda\Lambda of modules of a certain algebra B\mathscr{B} we will describe shortly. For the time being, only branes for which Lambda\Lambda is trivial will play a role for us.
5.3.3.
The space of morphisms between a pair of Lagrangian branes in a derived Fukaya category
is defined by Floer theory, which itself is modeled after Morse theory approach to supersymmetric quantum mechanics, from the introduction. The role of the Morse complex is taken by the Floer complex.
For branes equipped with a trivial local system, the Floer complex
is a graded vector space spanned by the intersection points of the two Lagrangians, together with the action of a differential QQ. The complex is graded by the fermion number, which is the Maslov index, and the equivariant gradings, thanks to the fact WW is multivalued.
is generated by instantons. In Floer theory, the coefficient of P^(')\mathcal{P}^{\prime} in QPQ \mathcal{P} is obtained by "counting" holomorphic strips in YY with boundary on L_(0)L_{0} and L_(1)L_{1}, interpolating from P\mathscr{P} to P^(')\mathcal{P}^{\prime}, of Maslov index 1 and equivariant degree 0 . The cohomology of the resulting complex is Floer cohomology.
5.3.4.
A simplification in the present case is that, just as branes have a description in terms of the Riemann surface, so do their intersection points, as well as the maps between them.
The theory that results is a generalization of Heegard-Floer theory, which is associated to ^(L)g=gl_(1∣1){ }^{L} \mathfrak{g}=\mathfrak{g l}_{1 \mid 1} and categorifies the Alexander polynomial [63,64]. Heegard-Floer theory
thought of as a configuration space of fermions on the Riemann surface, as opposed to anyons for Y_(Su_(2))Y_{\mathfrak{S u}_{2}}; in particular, their top holomorphic forms differ.
While we so far assumed that ^(L)g{ }^{L} \mathrm{~g} is simply laced, the D_(Y)\mathscr{D}_{Y} has an extension to nonsimply-laced Lie algebras, as well as gl_(m∣n)\mathrm{gl}_{m \mid n} and spp_(m∣2n)\mathfrak{s p} \mathfrak{p}_{m \mid 2 n} Lie superalgebras, described in [3,5].
5.4. Link invariants and equivariant mirror symmetry
Mirror symmetry helps us understand exactly which questions we need to ask to recover homological knot invariants from YY.
5.4.1.
Since YY is the ordinary mirror of XX, we should start by understanding how to recover homological knot invariants from XX, rather than X\mathcal{X}. Every B-brane on X\mathcal{X} which is relevant for us comes from a BB-brane on XX via an exact functor
{:(5.8)f_(**):D_(X)rarrDx:}\begin{equation*}
f_{*}: \mathscr{D}_{X} \rightarrow \mathscr{D} x \tag{5.8}
\end{equation*}
which interprets a sheaf "downstairs" on XX as a sheaf "upstairs" on X\mathcal{X}. The functor f_(**)f_{*} is more precisely the right-derived functor Rf_(**)R f_{*}. Its adjoint
is the left derived functor Lf^(**)L f^{*}, and corresponds to tensoring with the structure sheaf oxO_(X)\otimes \mathcal{O}_{X}, and restricting. Adjointness implies that, given any pair of branes on X\mathcal{X} that come from XX,
F=f_(**)F,quadE=f_(**)G\mathscr{F}=f_{*} F, \quad \mathcal{E}=f_{*} G
the Hom's upstairs, in D_(X)\mathscr{D}_{X}, agree with the Hom's downstairs, in D_(X)\mathscr{D}_{X},
after replacing FF with f^(**)f_(**)Ff^{*} f_{*} F. The functor f^(**)f_(**)f^{*} f_{*} is not identity on D_(X)\mathscr{D}_{X}.
5.4.2.
The equivariant homological mirror symmetry relating D_(X)\mathscr{D}_{X} and D_(Y)\mathscr{D}_{Y} is not an equivalence of categories, but a correspondence of branes and Hom's which come from a pair of adjoint functors h_(**)h_{*} and h^(**)h^{*}, inherited from f_(**)f_{*} and f^(**)f^{*} via the downstairs homological mirror symmetry:
Alternatively, h^(**)h^{*} and h_(**)h_{*} come by composing the upstairs mirror symmetry with a pair of functors k^(**):Dy rarrD_(Y)k^{*}: \mathscr{D} y \rightarrow \mathscr{D}_{Y} and k_(**):D_(Y)rarrDyk_{*}: \mathscr{D}_{Y} \rightarrow \mathscr{D} y, which are mirror to f^(**)f^{*} and f_(**)f_{*}. The functors k^(**),k_(**)k^{*}, k_{*} come from Lagrangian correspondences; their construction is described in joint work with McBreen, Shende, and Zhou [6]. The functor k_(**)k_{*} amounts to pairing a brane downstairs, with a vanishing torus fiber over it; this is how Figure 6 arises in our model example. The adjoint functors let us recover answers to all interesting questions about X\mathcal{X} from YY.
5.4.3.
For any simply laced Lie algebra ^(L)g{ }^{L} \mathrm{~g}, the branes U inDXU \in \mathscr{D} X which serve as cups upstairs are the structure sheaves of (products of) minuscule Grassmannians, as described in Section 4.3.6. They come via the functor h_(**)h_{*} from branes IU inD_(Y)I U \in \mathscr{D}_{Y} downstairs, on YY
U=h_(**)IUU=h_{*} I U
which are (products of) generalized intervals. A minuscule Grassmannian G//P_(i)G / P_{i} is the h_(**^(-))h_{*^{-}} image of a brane which is the configuration space of colored points on an interval ending on a pair of punctures on A\mathscr{A} corresponding to representations V_(i)V_{i} and V_(i)^(**)V_{i}^{*}. The points are colored by simple positive roots in mu_(i)+mu_(i)^(**)=sum_(a)d_(a,i)^(L)e_(a)\mu_{i}+\mu_{i}^{*}=\sum_{a} d_{a, i}{ }^{L} e_{a}, and ordered in the sequence by which, to obtain the lowest weight mu_(i)^(**)\mu_{i}^{*} in representation V_(i)V_{i}, we subtract simple positive
FIGURE 7
The cup and cap A-branes corresponding to the defining representation of ^(L)g=su_(4){ }^{L} \mathfrak{g}=\mathfrak{s u}_{4}, colored by its three simple roots; they are equivariant mirror to a B-brane supported on a P^(4)\mathbb{P}^{4} as its structure sheaf.
roots from the highest weight mu_(i)\mu_{i}. Because V_(i)V_{i} and V_(i)^(**)V_{i}^{*} are minuscule representations, the ordering and hence the brane IUI U is unique, up to equivalence and a choice of grading. The U\mathcal{U} branes project back down as generalized figure-eight branes; these are nested products of figure-eights, colored by simple roots
h^(**)U=h^(**)h_(**)IU=EUh^{*} U=h^{*} h_{*} I U=E U
and ordered analogously, as in Figure 7. As objects of D_(Y)\mathscr{D}_{Y}, these branes are best described iterated cones over more elementary branes, mirror to stable basis branes [5]. The cup and cap branes all come with trivial local systems, for which the Floer complexes are the familiar ones, given by (5.7).
As an example, for ^(L)g=SH_(2){ }^{L} \mathfrak{g}=\mathfrak{S H}_{2} the only minuscule representation is the defining representation V_(i)=V_((1)/(2))V_{i}=V_{\frac{1}{2}}, which is self-conjugate. The cup brane UU in X\mathcal{X} is a product of dd non-intersecting P^(1)\mathbb{P}^{1}, s. It comes, as the image of h_(**)h_{*}, from a brane IUI U in YY which is a product of dd simple intervals, connecting pairs of punctures that come together. The U\mathcal{U}-brane projects back down, via the h^(**)h^{*} functor, as a product of dd elementary figure-eight branes. The branes are graded by Maslov and equivariant gradings, as described in [2].
5.4.4.
In the description based on YY, both the branes, and the action of braiding on them is geometric, so we can simply start with a link and a choice of projection to the surface A=RxxS^(1)\mathcal{A}=\mathbb{R} \times S^{1}. A link contained in a three ball in R^(2)xxS^(1)\mathbb{R}^{2} \times S^{1} is equivalent to the same link in R^(3)\mathbb{R}^{3}, and projects to a contractible patch on A\mathcal{A}.
To translate the link to a pair of A-branes, start by choosing bicoloring of the link projection, such that each of its components has an equal number of red and blue segments, and the red always underpass the blue. For a link component colored by a representation V_(i)V_{i} of ^(L)g{ }^{L} \mathrm{~g}, place a puncture colored by its highest weight mu_(i)\mu_{i} where a blue segment begins and its conjugate mu_(i)^(**)\mu_{i}^{*} where it ends; the orientation of the link component distinguishes the two. The mirror Lagrangians I_(u)I_{u} and BE_(U)\mathscr{B} E_{\mathcal{U}} are obtained by replacing all the blue segments by the interval branes, and all the red segments by figure-eight branes, related by equivariant mirror symmetry to minuscule Grassmannian branes. This data determines both YY and the branes on it we need. The variant of the second step, applicable for Lie superalgebras, is described in [5].
FIGURE 8
A bicoloring of the left-handed trefoil.
Equivariant mirror symmetry predicts that a homological link invariant is the space of morphisms
the cohomology of the Floer complex of the two branes. In what follows, will explain how to compute it.
FIGURE 9
The branes corresponding to the left-handed trefoil in ^(L)g=su_(2){ }^{L} \mathfrak{g}=\mathfrak{s u}_{2}. The knot was isotoped relative to Figure 8 .
5.4.5.
To evaluate the Euler characteristic of the homology in (5.11), one simply counts intersection points of Lagrangians, keeping track of gradings. For links in R^(3)\mathbb{R}^{3}, the equivariant grading in (5.11) collapses to a Z\mathbb{Z}-grading. The Euler characteristic becomes
{:(5.12)chi(BEu","Iu)=bigoplus_(PinBEU nn IU)(-1)^(M(P))q^(J(P)):}\begin{equation*}
\chi(\mathscr{B} E u, I u)=\bigoplus_{\mathcal{P} \in \mathscr{B} E \mathcal{U \cap I U}}(-1)^{M(\mathcal{P})} q^{J(\mathcal{P})} \tag{5.12}
\end{equation*}
where M(P)M(\mathscr{P}) and J(P)J(\mathscr{P}) are the Maslov and c^(0)c^{0}-grading of the point P\mathscr{P}; as in Heegard-Floer theory, there are purely combinatorial formulas for them [3,5]. Mirror symmetry implies that this is the U_(q)(^(L)(g))U_{\mathfrak{q}}\left({ }^{L} \mathrm{~g}\right) invariant of the link in R^(3)\mathbb{R}^{3}.
The fact that for ^(L)g=su_(2){ }^{L} \mathfrak{g}=\mathfrak{s u}_{2} the graded count of intersection points in (5.12) reproduces the Jones polynomial is a theorem of Bigelow [15], building on the work of Lawrence [56-58]. Bigelow also proved the statement for ^(L)g=su_(N){ }^{L} \mathfrak{g}=\mathfrak{s u}_{N} with links colored by the defining representation [16]. The equivariant homological mirror symmetry explains the origin of Bigelow's peculiar construction, and generalizes it to other U_(q)(^(L)(g))U_{\mathrm{q}}\left({ }^{L} \mathrm{~g}\right) link invariants. ^(1){ }^{1}
5.4.6.
The action of the differential QQ on the Floer complex, defined by counting holomorphic maps from a disk DD to YY with boundaries on the pair of Lagrangians, should have a reformulation [2] in terms of counting holomorphic curves embedded in D xxAD \times \mathcal{A} with certain properties, generalizing the cylindrical formulation of Heegard-Floer theory due to Lipshitz [59]. The curve must have a projection to DD as a d=sum_(a)d_(a)d=\sum_{a} d_{a}-fold cover, with branching only between components of one color, and a projection to A\mathscr{A} as a domain with boundaries on one-dimensional Lagrangians of matching colors. In addition, the potential WW must pull back to DD as a regular holomorphic function. Computing the action of QQ in this framework reduces to solving a sequence of well defined, but hard, problems in complex analysis in one dimension, which are applications of the Riemannian mapping theorem, similar to that in [63].
proven in [4]. A basic virtue of mirror symmetry is that it sums up holomorphic curve counts. In our case, it solves all the disk-counting problems required to find the action of the differential QQ on the Floer complex underlying (6.1).
6.1. The algebra of A-branes
As in the simplest examples of homological mirror symmetry, D_(X)\mathscr{D}_{X} and D_(Y)\mathscr{D}_{Y} are both generated by a finite set of branes.
6.1.1.
From perspective of YY, the generating set of branes come from products of real line Lagrangians on A\mathscr{A}, colored by d=sum_(a)d_(a)d=\sum_{a} d_{a} simple roots. The brane is unchanged by reorder ing a pair of its neighboring Lagrangian components, provided they are colored by roots
1 In [60], Bigelow's representation of the Jones polynomial, specialized to q=1q=1, was related to the Euler characteristic of symplectic Khovanov homology of [73].
which are not linked in the Dynkin diagram (:^(L)e_(a),^(L)e_(b):)=0\left\langle{ }^{L} e_{a},{ }^{L} e_{b}\right\rangle=0. It is also unchanged by passing a component colored by ^(L)e_(a){ }^{L} e_{a} across a puncture colored by a weight mu_(i)\mu_{i} with (:^(L)e_(a),mu_(i):)=0\left\langle{ }^{L} e_{a}, \mu_{i}\right\rangle=0. There is a generating brane
is the generator of D_(Y)\mathscr{D}_{Y} which is mirror to the tilting vector bundle on XX, which generates D_(X)\mathscr{D}_{X}. This generalizes the simplest example of mirror symmetry from Section 5.1. As before, we will be denoting branes on XX and on YY related by homological mirror symmetry by the same letter.
6.1.2.
A well known phenomenon in mirror symmetry is that it may introduce Lagrangians with an extra structure of a local system, a nontrivial flat U(1)U(1) bundle. The mirror of a structure sheaf of a generic point, in our model example of mirror symmetry from Section 5.1, is a Lagrangian of this sort.
Here, we find a generalization of this [4]. The pair of adjoint functors h_(**)h_{*} and h^(**)h^{*} that relate D_(Y)\mathscr{D}_{Y} with its equivariant mirror D_(X)\mathscr{D}_{X} equip each TT-brane with a vector bundle or, more precisely, with a local system of modules for a graded algebra B\mathscr{B}. The algebra is a product B=⨂_(a=1)^(rk)B_(d_(a))\mathscr{B}=\bigotimes_{a=1}^{r k} \mathscr{B}_{d_{a}}, where B_(d)\mathscr{B}_{d} has a representation as the quotient of the algebra of polynomials in dd variables z_(1),dots,z_(d)z_{1}, \ldots, z_{d} which sets their symmetric functions to zero. The zz 's have equivariant qq-degree equal to one.
As a consequence, the downstairs theory is not simply the Fukaya category of YY, but a Fukaya category with coefficients [4]: Floer complexes assign to each intersection point PinL_(0)nnL_(1)\mathscr{P} \in L_{0} \cap L_{1} a vector space hom _(B)(Lambda_(0)|P,Lambda_(1)|P)_{\mathscr{B}}\left(\Lambda_{0}\left|\mathcal{P}, \Lambda_{1}\right| \mathcal{P}\right) of homomorphisms of B\mathscr{B}-modules Lambda_(0,1)\Lambda_{0,1} which L_(0,1)L_{0,1} are equipped with. The cup and cap branes E_(U)E_{\mathcal{U}} and IUI \mathcal{U} come with trivial modules for B\mathscr{B}. The T_(C)T_{\mathscr{C}} branes correspond to modules for B\mathscr{B} equal to B\mathscr{B} itself.
6.1.3.
Since the T_(C)T_{\mathcal{C}}-branes are noncompact, defining the Hom's between them requires care. The Hom's
are defined through a perturbation of T_(C)T_{\mathcal{C}} which induces wrapping near infinities of A\mathcal{A}, as in Figure 4, and other examples of wrapped Fukaya categories.
The Floer cohomology groups HF are cohomology groups of the Floer complex whose generators are intersection points of the T_(e)T_{e} branes, with coefficients in B\mathscr{B}. The generators all have homological degree zero, so the Floer differential is trivial, and
{:(6.3)Hom_(D_(Y))(T_(e),T_(e^('))[k]{( vec(d))})=0","quad" for all "k!=0" and all " vec(d):}\begin{equation*}
\operatorname{Hom}_{\mathscr{D}_{Y}}\left(T_{e}, T_{\mathcal{e}^{\prime}}[k]\{\vec{d}\}\right)=0, \quad \text { for all } k \neq 0 \text { and all } \vec{d} \tag{6.3}
\end{equation*}
into an algebra, which is an ordinary associative algebra, graded only by equivariant degrees.
6.1.4.
The vanishing in (6.3) mirrors the tilting property of TT viewed as the generator of D_(X)\mathscr{D}_{X}. The tilting vector bundle T inD_(X)T \in \mathscr{D}_{X} is inherited from the Bezrukavnikov-Kaledin tilting bundle T\mathcal{T} on X\mathcal{X},
T=bigoplus_(C)T_(e)inDx\mathcal{T}=\bigoplus_{\mathcal{C}} \mathcal{T}_{\mathcal{e}} \in \mathscr{D} x
from Section 4.4, as the image of the f^(**)f^{*} functor, which is tensoring with the structure sheaf of XX and restriction, f^(**)T=T inD_(X)f^{*} \mathcal{T}=T \in \mathscr{D}_{X}. The endomorphism of the upstairs tilting generator T ,
Since T\mathcal{T} is a vector bundle on X\mathcal{X}, the center of A\mathscr{A} is the algebra of holomorphic functions on X\mathcal{X}. The downstairs algebra is a quotient of the upstairs one, by a two-sided ideal
The ideal I\mathscr{I} is generated by holomorphic functions that vanish on the core XX.
6.1.5.
The cKLRW algebra A\mathscr{A} is defined as an algebra of colored strands on a cylinder, decorated with dots, where composition is represented by stacking cylinders and rescaling [80]. The local algebra relations are those of the ordinary KLRW algebra from [78]. Placing the theory on the cylinder, it gets additional gradings by the winding number of strands of a given color, corresponding to the equivariant Lambda\Lambda-action on X\mathcal{X}.
The elements of the algebra A=A//IA=\mathscr{A} / \mathscr{I} have a geometric interpretation by recalling the Floer complex CF^(**)(T_(C),T_(⨀'))\mathrm{CF}^{*}\left(T_{\mathcal{C}}, T_{\bigodot^{\prime}}\right) is fundamentally generated by paths rather the intersection points. The S^(1)S^{1} of the algebra cylinder is the S^(1)S^{1} in the Riemann surface A\mathcal{A}; its vertical direction parameterizes the path. The geometric intersection points of the TT-branes on A\mathscr{A} correspond to strings in AA. The flat bundle morphisms, a copy of B\mathscr{B} for each geometric intersection point, dress the strings by dots of the same color. The algebra B\mathscr{B} is the quotient, of the subalgebra of A\mathscr{A} generated by dots, by the ideal I\mathscr{I} of their symmetric functions.
6.2. The meaning of link homology
Since T=bigoplus_(ã¨)T_(e)T=\bigoplus_{ã¨} T_{\mathscr{e}} generates D_(Y)\mathscr{D}_{Y}, like every Lagrangian in D_(Y)\mathscr{D}_{Y}, the BEU\mathscr{B} E \mathcal{U} brane has a description as a complex
every term of which is a direct sum of T_(C)T_{\mathscr{C}}-branes. The complex is the projective resolution of the BE_(U)\mathscr{B} E_{U} brane. It describes how to get the BE_(U)inD_(Y)\mathscr{B} E_{U} \in \mathscr{D}_{Y} brane by starting with the direct sum brane
as a cohomological degree 1 and equivariant degree 0 operator, which squares to zero Q_(A)^(2)=0Q_{A}^{2}=0 in the algebra AA.
6.2.1.
The category of A-branes D_(Y)\mathscr{D}_{Y} has a second, Koszul dual set of generators, which are the vanishing cycle branes I=bigoplus_(↼)I_(↼)I=\bigoplus_{\leftharpoonup} I_{\leftharpoonup} of [2]. The II - and the TT-branes are dual in the sense that the only nonvanishing morphisms from the TT - to the II-branes are
The I_(C)I_{\mathscr{C}}-branes and the TCT \mathcal{C}-branes are, respectively, the simple and the projective modules of the algebra AA.
6.2.2.
Among the II-branes, we find the branes Iu inD_(Y)I u \in \mathscr{D}_{Y} which serve as cups. This is a wonderful simplification because it implies that from the complex in (6.5), we get for free a complex of vector spaces:
{:(6.9)0rarrhom_(A)(BE_(0)(T),Iu{( vec(d))})rarr"t_(0)"hom_(A)(BE_(1)(T),Iu{( vec(d))})rarr"t_(1)"cdots:}\begin{equation*}
0 \rightarrow \operatorname{hom}_{A}\left(\mathscr{B} E_{0}(T), I u\{\vec{d}\}\right) \xrightarrow{t_{0}} \operatorname{hom}_{A}\left(\mathscr{B} E_{1}(T), I u\{\vec{d}\}\right) \xrightarrow{t_{1}} \cdots \tag{6.9}
\end{equation*}
The maps in the complex (6.9) are induced from the complex in (6.5). The cohomologies of this complex are the link homologies we are after,
We learn that link homology captures only a small part of the geometry of BEU\mathscr{B} E \mathcal{U}, the braided cup brane, or more precisely, of the complex that resolves it. Because the TT branes are dual to the II-branes by (6.8), almost all terms in the complex (6.9) vanish. The cohomology (6.10) of small complex that remains is the U_(q)(^(L)(g))U_{\mathrm{q}}\left({ }^{L} \mathrm{~g}\right) link homology.
6.2.4.
The complex (6.9) itself has a geometric interpretation as the Floer complex,
CF^(**,**)(BEU,IU)\mathrm{CF}^{*, *}(\mathscr{B} E \mathcal{U}, I U)
Namely, the vector space at the kk th term of the complex
is identified, by construction described in section 6.3, with that spanned by the intersection points of the BEu\mathscr{B} E u brane and the IUI U brane, of cohomological degree [k][k] and equivariant degree { vec(d)}\{\vec{d}\}.
encode the action of the Floer differential. A priori, computing these requires counting holomorphic disk instantons. In our case, mirror symmetry (6.2) has summed them up.
6.3. Projective resolutions from geometry
The projective resolution in (6.5) encodes all the U_(q)(^(L)g)U_{\mathcal{q}}\left({ }^{L} \mathfrak{g}\right) link homology, and more. Finding the resolution requires solving two problems, both in general difficult. We will solve simultaneously [5][5].
6.3.1.
The first problem is to compute which module of the algebra AA the brane BEu\mathscr{B} E u gets mapped to by the Yoneda functor
L inD_(Y)rarrHom_(D_(Y))^(**,**)(T,L)inD_(A)L \in \mathscr{D}_{Y} \rightarrow \operatorname{Hom}_{\mathscr{D}_{Y}}^{*, *}(T, L) \in \mathscr{D}_{A}
This functor, which is the source of the equivalence D_(Y)~=D_(A)\mathscr{D}_{Y} \cong \mathscr{D}_{A}, maps a brane LL to a right module for AA, on which the algebra acts as
Evaluating this action requires counting disk instantons.
6.3.2.
The second problem is to find the resolution of this module, as in (6.5). The Yoneda functor maps the T_(e)T_{e} branes to projective modules of the algebra AA, so the resolution in (6.5) is a projective resolution of the AA module corresponding to the BE_(U)\mathscr{B} E_{\mathcal{U}} brane. This problem is known to be solvable, however, only formally so, by infinite bar resolutions.
6.3.3.
In our setting, these two problems get solved together. Fortune smiles since the BE_(U)inD_(Y)\mathscr{B} E_{\mathcal{U}} \in \mathscr{D}_{Y} branes are products of dd one-dimensional Lagrangians on A\mathcal{A}, for which the complex resolving brane (6.5) can be deduced explicitly, from the geometry of the brane.
6.3.4.
Take a pair of branes L^(')L^{\prime} and L^('')L^{\prime \prime} on YY which are products of dd one-dimensional Lagrangians on A\mathcal{A}, chosen to coincide up to one of their factors. Up to permutation, we can take
If L_(1)^(')L_{1}^{\prime} and L_(1)^('')L_{1}^{\prime \prime} (which are necessarily of the same color) intersect over a point p inL_(1)^(')nnL_(1)^('')p \in L_{1}^{\prime} \cap L_{1}^{\prime \prime} of Maslov index zero, we get a new one dimensional Lagrangian L_(1)L_{1} which is a cone over pp,
and which also has Maslov index zero, L=Cone(P)L=\operatorname{Cone}(\mathcal{P}).
Conversely, any LL brane which is of the product form in (6.11) can be written as a complex [11][11]
{:(6.13)L~=L^(')rarr"P"L^(''):}\begin{equation*}
L \cong L^{\prime} \xrightarrow{\mathcal{P}} L^{\prime \prime} \tag{6.13}
\end{equation*}
with an explicit map P\mathcal{P} coming from a one-dimensional intersection point in one of its factors, as in (6.12).
6.3.5.
To find the projective resolution of the BEU\mathscr{B} E \mathcal{U} brane in (6.5), start by isotoping the brane, by stretching it straight along the cylinder.
Let the brane break at the two infinities of A\mathscr{A}, to get the direct sum brane BE(T)\mathscr{B} E(T) in (6.6), on which the complex is based. To find the maps in the complex, record how the brane breaks, iterating the above construction, one one-dimensional intersection point at the time. Every intersection point of the form (6.12) translates to a specific element of the algebra AA and a specific map in the complex. The result is a product of dd one-dimensional complexes, which describes factors of BEU\mathscr{B} E \mathcal{U}, and captures almost all the terms in the differential Q_(A)Q_{A}. The remaining ones follow, up to quasi-isomorphisms, by asking that the differential closes Q_(A)^(2)=0Q_{A}^{2}=0 in the algebra AA. The fact that not all terms in Q_(A)Q_{A} are geometric is a general feature of d > 1d>1 theories.
In practice, it is convenient to first break the brane one of the two infinities of A\mathcal{A}, and only then on the other. The branes at the intermediate stage are images, under the h^(**)h^{*} functor, of stable basis branes [7,61][7,61] on DX\mathscr{D} X. The stable basis branes play a similar role to that of Verma modules in category O\mathcal{O}. The detailed algorithm is given in [5].
6.3.6.
As an example, take the left-handed trefoil and ^(L)g=su_(2){ }^{L} \mathfrak{g}=\mathfrak{s u}_{2}, which leads to the brane configuration from Figure 9. For simplicity, consider the reduced knot homology, where the unknot homology is set to be trivial. As in Heegard-Floer theory, this corresponds to erasing a component from the BEu\mathscr{B} E u and the IuI u branes, and leads to Figure 10. This also brings us back to the setting of our running example, where YY is the equivariant mirror to X\mathcal{X}, the resolution of the A_(n-1)A_{n-1} surface singularity, with n=4n=4.
FIGURE 10
Resolution of the BEU\mathscr{B} E \mathcal{U} brane corresponding to the reduced trefoil. The axis of the cylinder A\mathscr{A} is oriented vertically here; the branes do not wind around the S^(1)S^{1}.
The corresponding algebra A=bigoplus_(i,j=0)^(n-1)Hom_(D_(Y))^(**)(T_(i),T_(j))A=\bigoplus_{i, j=0}^{n-1} \operatorname{Hom}_{\mathscr{D}_{Y}}^{*}\left(T_{i}, T_{j}\right) is the path algebra of an affine A_(n-1)A_{n-1} quiver, whose nodes correspond to T_(i)T_{i} branes. The arrows a_(i+1,i)ina_{i+1, i} \inHom_(D_(Y))(T_(i),T_(i+1))\operatorname{Hom}_{\mathscr{D}_{Y}}\left(T_{i}, T_{i+1}\right) and b_(i,i+1)inHom_(DY)(T_(i+1),T_(i){1})b_{i, i+1} \in \operatorname{Hom}_{\mathscr{D} Y}\left(T_{i+1}, T_{i}\{1\}\right) satisfy a_(i,i-1)b_(i-1,i)=0=b_(i,i+1)a_(i+1,i)a_{i, i-1} b_{i-1, i}=0=b_{i, i+1} a_{i+1, i}, with ii defined modulo nn. The aa 's and bb 's correspond to intersections of TT-branes, near one or the other infinity of A\mathscr{A}; we have suppressed their Lambda\Lambda-equivariant degrees.
Isotope the BEu\mathscr{B} E \mathcal{u} brane straight along the cylinder A\mathcal{A}. Let it break into TT-branes, as in Figure 10, while recording how the brane breaks, one connected sum at a time. Every connected sum of a pair of TT-branes is a cone over their intersection point, at one of the two infinities of A\mathcal{A}, and a specific element of the algebra AA. This leads to the complex
which closes by the AA-algebra relations.
The reduced homology of the trefoil is the cohomology of the complex hom_(A)(BE^(∙),Iu{d})\operatorname{hom}_{A}\left(\mathscr{B} E^{\bullet}, \operatorname{I} u\{d\}\right) in (6.9). The only non-zero contributions come from the T_(2)T_{2} brane, since the cup brane IU=I_(2)I U=I_{2} is dual to it. All the maps evaluate to zero, as IUI U brane is a simple module for AA. As a consequence,
Hom_(D_(Y))(BEu,Iu[k]{d})=H^(k)(hom_(A)(BE^(∙),I_(2){d}))\operatorname{Hom}_{\mathscr{D _ { Y }}}(\mathscr{B} E u, I u[k]\{d\})=H^{k}\left(\operatorname{hom}_{A}\left(\mathscr{B} E^{\bullet}, I_{2}\{d\}\right)\right)
equals to Z\mathbb{Z} only for (k,d)=(0,0),(2,-2),(3,-3)(k, d)=(0,0),(2,-2),(3,-3), and vanishes otherwise. Here, k=Mk=M is the Maslov or cohomological degree and d=Jd=J the Jones grading. This is the reduced Khovanov homology of the left-handed trefoil, up to regrading: Khovanov's (i,j)(i, j) gradings are related to (M,J)(M, J) by i=M+2J+i_(0)i=M+2 J+i_{0} and j=2J+j_(0)j=2 J+j_{0} where i_(0)=0,j_(0)=d+n_(+)-i_{0}=0, j_{0}=d+n_{+}-n_(-)n_{-}, where n_(+)=0,n_(-)=3n_{+}=0, n_{-}=3 are the numbers of positive and negative crossings, and d=1d=1 is the dimension of YY [2].
6.3.7.
The theory extends to non-simply-laced Lie algebras, and to Lie superalgebras gl_(m∣n)\mathfrak{g l}_{m \mid n} and _(m∣2n)\mathfrak{}_{m \mid 2 n}, as described in [5]. The algebra AA corresponding to ^(L)g{ }^{L} \mathrm{~g} which is a Lie superalgebra, is not an ordinary associative algebra but a differential graded algebra; the projective resolutions are then in terms of twisted complexes. This section gives a method for solving the theory which is new even for ^(L)g=gl_(1∣1){ }^{L} \mathfrak{g}=\mathfrak{g l}_{1 \mid 1}, corresponding to Heegard-Floer theory. The solution differs from that in [65], in particular since our Heegard surface is A=RxxS^(1)\mathcal{A}=\mathbb{R} \times S^{1}, independent of the link.
ACKNOWLEDGMENTS
This work grew out of earlier collaborations with Andrei Okounkov, which were indispensable. It includes results obtained jointly with Ivan Danilenko, Elise LePage, Yixuan Li, Michael McBreen, Miroslav Rapcak, Vivek Shende, and Peng Zhou. I am grateful to all of them for collaboration. I was supported by the NSF foundation grant PHY1820912, by the Simons Investigator Award, and by the Berkeley Center for Theoretical Physics.
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MINA AGANAGIC
Department of Mathematics, University of California, Berkeley, USA and Center for Theoretical Physics, University of California, Berkeley, USA, aganagic @ berkeley.edu
VECTOR BUNDLES ON ALGEBRAIC VARIETIES
ARAVIND ASOK AND JEAN FASEL
ABSTRACT
We survey recent developments related to the problem of classifying vector bundles on algebraic varieties. We focus on the striking analogies between topology and algebraic geometry, and the way in which the Morel-Voevodsky motivic homotopy category can be used to exploit those analogies.
In the mid-1950s, Serre created a dictionary between the theory of vector bundles in topology and the theory of projective modules over a commutative ring [55, 56]. Echoing M. M. Postnikov's MathSciNet review of Serre's paper, J. F. Adams prosaically wrote in his review of H. Bass' paper [22]: "This leads to the following programme: take definitions, constructions and theorems from bundle-theory; express them as particular cases of definitions, constructions and statements about finitely-generated projective modules over a general ring; and finally, try to prove the statements under suitable assumptions". One of the results Serre presented to illustrate this dictionary was the algebro-geometric analog of existence of nowhere vanishing sections for negative corank projective modules, now frequently referred to as Serre's splitting theorem, which we recall in algebro-geometric formulation: if E\mathscr{E} is a rank rr vector bundle over a Noetherian affine scheme XX of dimension dd, then when r > d,E~=E^(')o+O_(X)r>d, \mathcal{E} \cong \mathcal{E}^{\prime} \oplus \mathcal{O}_{X}.
After the Pontryagin-Steenrod representability theorem, topological vector bundles on smooth manifolds (or spaces having the homotopy type of a CW complex) can be analyzed using homotopy theoretic techniques. Extending Serre's analogy further and using celebrated work of Bass, Quillen, Suslin, and Lindel, F. Morel showed that algebraic vector bundles on smooth affine varieties could be studied using an algebro-geometric homotopy theory: the Morel-Voevodsky motivic homotopy theory. In this note, we survey recent developments in the theory of algebraic vector bundles motivated by this circle of ideas, making sure to indicate the striking analogies between topology and algebraic geometry.
To give the reader a taste of the methods we will use, we mention two results here.
First, we state an improvement of Serre's splitting theorem mentioned above (for the moment it suffices to know that A^(1)\mathbb{A}^{1}-cohomological dimension is bounded above by Krull dimension, but can be strictly smaller). Second, we will discuss the splitting problem for projective modules in corank 1, which goes beyond any classical results.
Theorem 1.1. If kk is a field, and XX is a smooth affine kk-scheme of A^(1)\mathbb{A}^{1}-cohomological dimension <= d\leq d, then any rank r > dr>d bundle splits off a trivial rank 1 summand.
Conjecture 1.2. Assume kk is an algebraically closed field, and X=Spec RX=\operatorname{Spec} R is a smooth affine kk-variety of dimension dd. A rank d-1d-1 vector bundle &\mathcal{\&} on XX splits off a free rank 1 summand if and only if 0=c_(d-1)(E)inCH^(d-1)(X)0=c_{d-1}(\mathcal{E}) \in \mathrm{CH}^{d-1}(X).
In Theorem 4.12 we verify Conjecture 1.2 in case d=3,4d=3,4 (and kk has characteristic not equal to 2). To motivate the techniques used to establish these results, we begin by analyzing topological variants of these conjectures. We close this note with a discussion of joint work with Mike Hopkins which addresses the difficult problem of constructing interesting low rank vector bundles on "simple" algebraic varieties. As with any survey, this one reflects the biases and knowledge of the authors. Limitations of space have prevented us from talking about a number of very exciting and closely related topics.
2. A FEW TOPOLOGICAL STORIES
In this section, we recall a few topological constructions that elucidate the approaches we use to analyze corresponding algebro-geometric questions studied later.
2.1. Moore-Postnikov factorizations
Suppose f:E rarr Bf: E \rightarrow B is a morphism of pointed, connected topological spaces having the homotopy type of CW\mathrm{CW} complexes that induces an isomorphism of fundamental groups (for simplicity of discussion). Write FF for the "homotopy" fiber of ff, so that there is a fiber sequence
F rarr Erarr"f"BF \rightarrow E \xrightarrow{f} B
yielding a long exact sequence relating the homotopy of F,EF, E, and BB.
A basic question that arises repeatedly is the following: given a map M rarr BM \rightarrow B, when can it be lifted along ff to a map M rarr EM \rightarrow E ? To approach this problem, one method is to factor ff in such a way as to break the original lifting problem into simpler problems where existence of a lift can be checked by, say, cohomological means.
One systematic approach to analyzing this question was laid out in the work of Moore-Postnikov. In this case, one factors ff so as to build EE out of BB by sequentially adding higher homotopy of ff (keeping track of the induced action of pi_(1)(E)~=pi_(1)(B)\pi_{1}(E) \cong \pi_{1}(B) on the fiber). In more detail, the Moore-Postnikov tower of ff consists of a sequence of spaces tau_( <= i)f,i >= 0\tau_{\leq i} f, i \geq 0 and morphisms fitting into the following diagram:
The key properties of this factorization are that (i) the composite maps E rarrtau_( <= i)f rarr BE \rightarrow \tau_{\leq i} f \rightarrow B all coincide with ff, (ii) the maps E rarrtau_( <= i)fE \rightarrow \tau_{\leq i} f induce isomorphisms on homotopy groups in degrees <= i\leq i, (iii) the maps tau_( <= i)f rarr B\tau_{\leq i} f \rightarrow B induce isomorphisms on homotopy in degrees > i+1>i+1, and (iv) there is a homotopy pullback diagram of the form
In particular, the morphism tau_( <= i)f rarrtau_( <= i-1)f\tau_{\leq i} f \rightarrow \tau_{\leq i-1} f is a twisted principal fibration, which means that a morphism M rarrtau_( <= i-1)fM \rightarrow \tau_{\leq i-1} f lifts along the tower if and only if the composite M rarrK^(pi_(1)(E))(pi_(i)(F),i+1)M \rightarrow K^{\pi_{1}(E)}\left(\pi_{i}(F), i+1\right) lifts to Bpi_(1)(E)\mathrm{B} \pi_{1}(E). The latter map amounts to a cohomology class on MM with coefficients in a local coefficient system; this cohomology class is pulled back from a "universal example" the kk-invariant at the corresponding stage. If the obstruction vanishes, a lift exists. Lifts are not unique in general, but the ambiguity in choice of a lift can also be described.
2.2. The topological splitting problem
In this section, to motivate some of the algebro-geometric results we will describe later, we review the problem of deciding whether a bundle of corank 0 or 1 on a closed smooth manifold MM of dimension d+1d+1 has a nowhere vanishing section. We now phrase this problem as a lifting problem of the type described in the preceding section.
In this case, the relevant lifting problem is:
To analyze the lifting problem, we describe the Moore-Postnikov factorization of ff. The homotopy fiber of ff coincides with the standard sphere S^(d-1)~=O(d)//O(d-1)S^{d-1} \cong O(d) / O(d-1).
The stabilization map O(d-1)rarr O(d)O(d-1) \rightarrow O(d) is compatible with the determinant, and there are thus induced isomorphisms pi_(1)(BO(d-1))rarrpi_(1)(BO(d))~=Z//2\pi_{1}(\mathrm{~B} O(d-1)) \rightarrow \pi_{1}(\mathrm{~B} O(d)) \cong \mathbb{Z} / 2 compatible with ff. Note, however, that the action of Z//2\mathbb{Z} / 2 on the higher homotopy of BO(d)\mathrm{B} O(d) depends on the parity of dd : when dd is odd the action is trivial, while if dd is even the action is nontrivial in general and even fails to be nilpotent. Of course, S^(d-1)S^{d-1} is (d-2)(d-2)-connected.
Remark 2.1. At this stage, the fact that bundles of negative corank on spaces have the homotopy type of a CW complex of dimension dd follows immediately from obstruction theory granted the assertion that the sphere S^(r)S^{r} is an (r-1)(r-1)-connected space in conjunction with the fact that negative corank means r > dr>d.
In order to write down obstructions, we need some information about the homotopy of spheres: the first nonvanishing homotopy group of S^(d-1)S^{d-1} is pi_(d-1)(S^(d-1))\pi_{d-1}\left(S^{d-1}\right) which coincides
with Z\mathbb{Z} for all d >= 2d \geq 2 (via the degree map). Likewise, pi_(d)(S^(d-1))\pi_{d}\left(S^{d-1}\right) is Z\mathbb{Z} if d=3d=3 and Z//2\mathbb{Z} / 2 if d > 3d>3 and is generated by a suitable suspension of the classical Hopf map eta:S^(3)rarrS^(2)\eta: S^{3} \rightarrow S^{2}.
Assume now XX is a space having the homotopy type of a finite CW\mathrm{CW} complex of dimension d+1d+1 for some fixed integer d >= 2d \geq 2 (to eliminate some uninteresting cases) and xi:X rarrBO(d)\xi: X \rightarrow \mathrm{B} O(d) classifies a rank dd vector bundle on XX. The first nonzero kk-invariant for ff yields a map X rarrK^(Z//2)(pi_(d-1)(S^(d-1)),d)X \rightarrow K^{\mathbb{Z} / 2}\left(\pi_{d-1}\left(S^{d-1}\right), d\right), i.e., an element
called the (twisted) Euler class, where Z[sigma]\mathbb{Z}[\sigma] is Z\mathbb{Z} twisted by the orientation character sigma\sigma defined by applying pi_(1)\pi_{1} to the morphism X rarrBO(d)rarrB(Z//2)X \rightarrow \mathrm{B} O(d) \rightarrow \mathrm{B}(\mathbb{Z} / 2).
Assuming this primary obstruction vanishes, one may choose a lift to the next stage of the Postnikov tower. If we fix a lift, then there is a well-defined secondary obstruction to lifting, that comes from the next kk-invariant: this obstruction is given by a map X rarrK^(Z//2)(pi_(d)(S^(d-1)),d+1)X \rightarrow K^{\mathbb{Z} / 2}\left(\pi_{d}\left(S^{d-1}\right), d+1\right), i.e., a cohomology class in H^(d+1)(X,Z[sigma])H^{d+1}(X, \mathbb{Z}[\sigma]) if d=3d=3 or H^(d+1)(X,Z//2)H^{d+1}(X, \mathbb{Z} / 2) if d!=3d \neq 3; in the latter case the choice of orientation character no longer affects this cohomology group.
If one tracks the effect of choice of lift on the obstruction class described above, one obtains a map K^(Z//2)(pi_(d-1)(S^(d-1)),d-1)rarrK^(Z//2)(pi_(d)(S^(d-1)),d+1)K^{\mathbb{Z} / 2}\left(\pi_{d-1}\left(S^{d-1}\right), d-1\right) \rightarrow K^{\mathbb{Z} / 2}\left(\pi_{d}\left(S^{d-1}\right), d+1\right), which is a twisted cohomology operation. If d=3d=3, the map in question is a twisted version of the Pontryagin squaring operation, while if d > 3d>3 the operation can be described as Sq^(2)+w_(2)uu\mathrm{Sq}^{2}+w_{2} \cup, where w_(2)w_{2} is the second Stiefel-Whitney class of the bundle. In that case, the secondary obstruction yields a well-defined coset in
This description of the primary and secondary obstructions was laid out carefully by the early 1950s by S. D. Liao [37].
Finally, the dimension assumption on XX guarantees that a lift of xi\xi along ff exists if and only if these two obstructions vanish. In principle, this kind of analysis can be continued, though the calculations become more involved as the indeterminacy created by successive choices of lifts becomes harder to control and information about higher unstable homotopy of spheres is also harder to obtain. For a thorough treatment of this and even more general situations, we refer the reader to [61].
Remark 2.2. The analysis of the obstructions can be improved by organizing the calculations differently. The Moore-Postnikov factorization has the effect of factoring a map f:X rarr Yf: X \rightarrow Y as a tower of fibrations where the relevant fibers are Eilenberg-Mac Lane spaces. However, there are many other ways to produce factorizations of ff with different constraints on the "cohomological" properties of pieces of the tower.
3. A QUICK REVIEW OF MOTIVIC HOMOTOPY THEORY
Motivic homotopy theory, introduced by F. Morel and V. Voevodsky [41], provides a homotopy theory for schemes over a base. While there are a number of different approaches
to constructing the motivic homotopy category that work in great generality, we work in a very concrete situation. By an algebraic variety over a field kk, we will mean a separated, finite type, reduced kk-scheme. We write Sm_(k)\mathrm{Sm}_{k} for the category of smooth algebraic varieties; for later use, we will also write Sm_(k)^("aff ")\mathrm{Sm}_{k}^{\text {aff }} for the full subcategory of Sm_(k)\mathrm{Sm}_{k} consisting of affine schemes.
The category Sm_(k)\mathrm{Sm}_{k} is "too small" to do homotopy theory, in the sense that various natural categorical constructions one would like to make (increasing unions, quotients by subspaces, etc.) can leave the category. As such, one first enlarges Sm_(k)\mathrm{Sm}_{k} to a suitable category Spc_(k)\mathrm{Spc}_{k} of "spaces"; one may take Spc_(k)\mathrm{Spc}_{k} to be the category of simplicial presheaves on Sm_(k)\mathrm{Sm}_{k} and the functor Sm_(k)rarrSpc_(k)\mathrm{Sm}_{k} \rightarrow \mathrm{Spc}_{k} is given by the Yoneda embedding followed by the functor viewing a presheaf on Sm_(k)\mathrm{Sm}_{k} as a constant simplicial presheaf.
3.1. Homotopical sheaf theory
Passing to Spc_(k)\mathrm{Spc}_{k} has the effect of destroying certain colimits that one would like to retain. To recover the colimits that have been lost, one localizes Spc_(k)\mathrm{Spc}_{k} and passes to a suitable "local" homotopy category of the sort first studied in detail by K. Brown-S. Gersten, A. Joyal, and J.F. Jardine: one fixes a Grothendieck topology tau\tau on Sm_(k)\mathrm{Sm}_{k} and inverts the so-called tau\tau-local weak equivalences on Spc_(k)\mathrm{Spc}_{k}; we refer the reader to [34] for a textbook treatment. We write H_(tau)(k)\mathrm{H}_{\tau}(k) for the resulting localization of Spc_(k)\mathrm{Spc}_{k}. If XinSpc_(k)\mathscr{X} \in \mathrm{Spc}_{k}, then a base-point for X\mathscr{X} is a morphism x:Spec k rarrXx: \operatorname{Spec} k \rightarrow \mathscr{X} splitting the structure morphism. There is an associated pointed homotopy category and these homotopy categories can be thought of as providing a convenient framework for "nonabelian" homological algebra.
In the category of pointed spaces, we can make sense of wedge sums and smash products, just as in ordinary topology. We also define spheres S^(i),i >= 0S^{i}, i \geq 0, as the constant simplicial presheaves corresponding to the simplicial sets S^(i)S^{i}. For any pointed space (X,x)(\mathscr{X}, x), we define its homotopy sheaves pi_(i)(X,x)\pi_{i}(\mathscr{X}, x) as the Nisnevich sheaves associated with the presheaves on Sm_(k)\mathrm{Sm}_{k} defined by
here the subscript + means adjoint a disjoint base-point. These homotopy sheaves may be used to formulate a Whitehead theorem.
If G\mathbf{G} is a Nisnevich sheaf of groups on Sm_(k)\mathrm{Sm}_{k}, then there is a classifying space B G such that for any smooth kk-scheme XX one has a functorial identification of pointed sets of the form
where we as usual identify isomorphism classes of rank n\operatorname{rank} n vector bundles locally trivial with respect to the Zariski topology on XX with GL_(n)\mathrm{GL}_{n}-torsors (and the choice of topology does not matter).
If A\mathbf{A} is any Nisnevich sheaf of abelian groups on Sm_(k)\mathrm{Sm}_{k}, then for any integer n >= 0n \geq 0 there are Eilenberg-Mac Lane spaces K(A,n)\mathrm{K}(\mathbf{A}, n), i.e., spaces with exactly one nonvanishing homotopy sheaf, appearing in degree nn, isomorphic to A\mathbf{A}. For such spaces, hom_(H_("Nis ")(k))(X,K(A,n))\operatorname{hom}_{\mathrm{H}_{\text {Nis }}(k)}(X, \mathrm{~K}(\mathbf{A}, n)) has a natural abelian group structure, and there are functorial isomorphisms of abelian groups
With this definition, for essentially formal reasons there is a suspension isomorphism for Nisnevich cohomology with respect to the suspension S^(1)^^(-)S^{1} \wedge(-).
3.2. The motivic homotopy category
The motivic homotopy category is obtained as a further localization of H_(Nis)(k)\mathrm{H}_{\mathrm{Nis}}(k) : one localizes at the projection morphisms XxxA^(1)rarrX\mathscr{X} \times \mathbb{A}^{1} \rightarrow \mathscr{X}. We write H(k)\mathrm{H}(k) for the resulting homotopy category; isomorphisms in this category will be referred to as A^(1)\mathbb{A}^{1}-weak equivalences. Following the notation in classical homotopy theory, we write
and refer to this set as the set of A^(1)\mathbb{A}^{1}-homotopy classes of maps from X\mathscr{X} to Y\mathscr{Y}.
If X\mathscr{X} is a space, we will write pi_(0)^(A^(1))(X)\pi_{0}^{\mathbb{A}^{1}}(\mathscr{X}) for the Nisnevich sheaf associated with the presheaf U|->[U,X]_(A^(1))U \mapsto[U, \mathscr{X}]_{\mathbb{A}^{1}} on Sm_(k)\operatorname{Sm}_{k}; we refer to pi_(0)^(A^(1))(X)\pi_{0}^{\mathbb{A}^{1}}(\mathscr{X}) as the sheaf of connected components, and we say that X\mathscr{X} is A^(1)\mathbb{A}^{1}-connected if pi_(0)^(A^(1))(X)\pi_{0}^{\mathbb{A}^{1}}(\mathscr{X}) is the sheaf Spec(k)\operatorname{Spec}(k).
We consider G_(m)\mathbb{G}_{m} as a pointed space, with base point its identity section 1 . In that case, we define motivic spheres
We caution the reader that there are a number of different indexing conventions used for motivic spheres. One defines bigraded homotopy sheaves pi_(i,j)^(A^(1))(X,x)\pi_{i, j}^{\mathbb{A}^{1}}(\mathscr{X}, x) for any pointed space as the Nisnevich sheaves associated with the presheaves on Sm_(k)\mathrm{Sm}_{k}
we write pi_(i)^(A^(1))(X,x)\pi_{i}^{\mathbb{A}^{1}}(\mathscr{X}, x) for pi_(i,0)^(A^(1))(X)\pi_{i, 0}^{\mathbb{A}^{1}}(\mathscr{X}). We will say that a pointed space (X,x)(\mathscr{X}, x) is A^(1)\mathbb{A}^{1} - kk-connected for some integer k >= 1k \geq 1 if it is A^(1)\mathbb{A}^{1}-connected and the sheaves pi_(i)^(A^(1))(X,x)\pi_{i}^{\mathbb{A}^{1}}(\mathscr{X}, x) are trivial for 1 <= i <= k1 \leq i \leq k. Because of the form of the Whitehead theorem in the Nisnevich local homotopy category, the sheaves pi_(i)^(A^(1))(-)\pi_{i}^{\mathbb{A}^{1}}(-) detect A^(1)\mathbb{A}^{1}-weak equivalences.
We write Delta_(k)^(∙)\Delta_{k}^{\bullet} for the cosimplicial affine space with
For any space X\mathscr{X}, we write Sing^(A^(1))X\operatorname{Sing}^{\mathbb{A}^{1}} \mathscr{X} for the space diag hom__(Delta^(∙),X)\operatorname{diag} \underline{\underline{h o m}}\left(\Delta^{\bullet}, \mathscr{X}\right). There is a canonical map XrarrSing^(A^(1))X\mathscr{X} \rightarrow \operatorname{Sing}^{\mathbb{A}^{1}} \mathscr{X} and the space Sing^(A^(1))X\operatorname{Sing}^{\mathbb{A}^{1}} \mathscr{X} is called the singular construction on X\mathscr{X}. For
a smooth scheme UU, the set of connected components pi_(0)(Sing^(A^(1))X(U))\pi_{0}\left(\operatorname{Sing}^{\mathbb{A}^{1}} \mathscr{X}(U)\right) will be called the set of naive A^(1)\mathbb{A}^{1}-homotopy classes of maps U rarrXU \rightarrow \mathscr{X} (by construction, it is the quotient of the set of morphisms U rarrXU \rightarrow \mathscr{X} by the equivalence relation generated by maps U xxA^(1)rarrXU \times \mathbb{A}^{1} \rightarrow \mathscr{X} ). Again, by definition there is a comparison morphism
{:(3.1)pi_(0)(" Sing "^(A^(1))X(U))rarr[U","X]_(A^(1)):}\begin{equation*}
\pi_{0}\left(\text { Sing }^{\mathbb{A}^{1}} \mathscr{X}(U)\right) \rightarrow[U, \mathscr{X}]_{\mathbb{A}^{1}} \tag{3.1}
\end{equation*}
Typically, the map (3.1) is far from being a bijection.
3.3. A^(1)\mathbb{A}^{1}-weak equivalences
We now give a number of examples of A^(1)\mathbb{A}^{1}-weak equivalences, highlighting some examples and constructions that will be important in the sequel.
Example 3.1. A smooth kk-scheme XX is called A^(1)\mathbb{A}^{1}-contractible if the structure morphism X rarr Spec kX \rightarrow \operatorname{Spec} k is an A^(1)\mathbb{A}^{1}-weak equivalence. By construction, A^(n)\mathbb{A}^{n} is an A^(1)\mathbb{A}^{1}-contractible smooth kk-scheme. However, there are a plethora of A^(1)\mathbb{A}^{1}-contractible smooth kk-schemes that are nonisomorphic to A^(n)\mathbb{A}^{n}. For instance, the Russell cubic threefold, defined by the hypersurface equation x+x^(2)y+z^(2)+t^(3)=0x+x^{2} y+z^{2}+t^{3}=0 is known to be nonisomorphic to affine space and also A^(1)\mathbb{A}^{1}-contractible [28]. See [16] for a survey of further examples.
Example 3.2. If f:X rarr Yf: X \rightarrow Y is a Nisnevich locally trivial morphism with fibers that are A^(1)\mathbb{A}^{1}-contractible smooth kk-schemes, then ff is an A^(1)\mathbb{A}^{1}-weak equivalence. Thus, the projection morphism for a vector bundle is an A^(1)\mathbb{A}^{1}-weak equivalence. A vector bundle EE over a scheme XX can be seen as a commutative algebraic XX-group scheme, so we may speak of EE-torsors; EE-torsors are classified by the coherent cohomology group H^(1)(X,E)\mathrm{H}^{1}(X, \mathscr{E}) (in particular, vector bundle torsors over affine schemes may always be trivialized). Vector bundle torsors are Zariski locally trivial fiber bundles with fibers isomorphic to affine spaces, and the projection morphism for a vector bundle torsor is an A^(1)\mathbb{A}^{1}-weak equivalence.
By an affine vector bundle torsor over a scheme XX we will mean a torsor pi:Y rarr X\pi: Y \rightarrow X for some vector bundle EE on XX such that YY is an affine scheme. Jouanolou proved [35, LEMME 1.5] that any quasiprojective variety admits an affine vector bundle torsor. Thomason [63, PROPOSITION 4.4] generalized Jouanolou's observation, and the following result is a special case of his results.
Lemma 3.3 (Jouanolou-Thomason homotopy lemma). If XX is a smooth kk-variety, then XX admits an affine vector bundle torsor. In particular, any smooth kk-variety is isomorphic in H(k)\mathrm{H}(k) to a smooth affine variety.
Definition 3.4. By a Jouanolou device for a smooth kk-variety XX we will mean a choice of an affine vector bundle torsor p:Y rarr Xp: Y \rightarrow X.
Example 3.5. When X=P^(n)X=\mathbb{P}^{n} there is a very simple construction of a "standard" Jouanolou device tilde(P)^(n)\tilde{\mathbb{P}}^{n}. Geometrically, the standard Jouanolou device for P^(n)\mathbb{P}^{n} may be described as the complement of the incidence divisor in P^(n)xxP^(n)\mathbb{P}^{n} \times \mathbb{P}^{n} where the second projective space is viewed as the dual of the first, with structure morphism the projection onto either factor.
Example 3.6. If XX is a smooth projective variety of dimension dd, then we may choose a finite morphism psi:X rarrP^(d)\psi: X \rightarrow \mathbb{P}^{d}. Pulling back the standard Jouanolou device for P^(d)\mathbb{P}^{d} along psi\psi, we see that XX admits a Jouanolou device tilde(X)\tilde{X} of dimension 2d2 d.
Example 3.7. For n inNn \in \mathbb{N}, consider the smooth affine kk-scheme Q_(2n-1)Q_{2 n-1} defined as the hypersurface in A_(k)^(2n)\mathbb{A}_{k}^{2 n} given by the equation sum_(i=1)^(n)x_(i)y_(i)=1\sum_{i=1}^{n} x_{i} y_{i}=1. Projecting onto the first nn-factors, we obtain a map p:Q_(2n-1)rarrA^(n)\\0p: Q_{2 n-1} \rightarrow \mathbb{A}^{n} \backslash 0 which one may check is an affine vector bundle torsor. For any integer n >= 0,A^(n)\\0n \geq 0, \mathbb{A}^{n} \backslash 0 is A^(1)\mathbb{A}^{1}-weakly equivalent to S^(n-1,n)S^{n-1, n} (see [41, §3.2, EXAMPLE 2.20]) and consequently Q_(2n-1)Q_{2 n-1} is A^(1)\mathbb{A}^{1}-weakly equivalent to S^(n-1,n)S^{n-1, n} as well.
Example 3.8. For n inNn \in \mathbb{N}, consider the smooth affine kk-scheme Q_(2n)Q_{2 n} defined as the hypersurface in A_(k)^(2n+1)\mathbb{A}_{k}^{2 n+1} given by the equation
The variety Q_(2)Q_{2} is isomorphic to the standard Jouanolou device over P^(1)\mathbb{P}^{1}. The variety P^(1)\mathbb{P}^{1} is A^(1)\mathbb{A}^{1}-weakly equivalent to S^(1,1)S^{1,1} and therefore Q_(2)Q_{2} is A^(1)\mathbb{A}^{1}-weakly equivalent to S^(1,1)S^{1,1} as well. For n >= 2n \geq 2, one knows that Q_(2n)Q_{2 n} is A^(1)\mathbb{A}^{1}-weakly equivalent to S^(n,n)S^{n, n} [2, THEOREM 2].
3.4. Representability results
If F\mathscr{F} is a presheaf on Sm_(k)\mathrm{Sm}_{k}, we will say that F\mathscr{F} is A^(1)\mathbb{A}^{1}-invariant (resp. A^(1)\mathbb{A}^{1}-invariant on affines) if the pullback map F(X)rarrF(X xxA^(1))\mathscr{F}(X) \rightarrow \mathscr{F}\left(X \times \mathbb{A}^{1}\right) is an isomorphism for all X inSm_(k)X \in \operatorname{Sm}_{k} (resp. X inSm_(k)^("aff ")X \in \mathrm{Sm}_{k}^{\text {aff }} ). A necessary condition for a cohomology theory on smooth schemes to be representable in H(k)\mathrm{H}(k) is that it is A^(1)\mathbb{A}^{1}-invariant and has a Mayer-Vietoris property with respect to the Nisnevich topology. One of the first functors that one encounters with these properties is that which assigns to a smooth kk-scheme its Picard group. Morel and Voevodsky showed [41, $4 PROPOSITION 3.8] that if XX is a smooth kk-scheme, then the A^(1)\mathbb{A}^{1}-weak equivalence P^(oo)rarrBG_(m)\mathbb{P}^{\infty} \rightarrow \mathrm{B} \mathbb{G}_{m} induces a bijection [X,P^(oo)]_(A^(1))~=Pic(X)\left[X, \mathbb{P}^{\infty}\right]_{\mathbb{A}^{1}} \cong \operatorname{Pic}(X).
If A\mathbf{A} is a sheaf of abelian groups on Sm_(k)\mathrm{Sm}_{k}, then the functors H_(Nis)^(i)(-,A)\mathrm{H}_{\mathrm{Nis}}^{i}(-, \mathbf{A}) frequently fail to be A^(1)\mathbb{A}^{1}-invariant (taking A=G_(a)\mathbf{A}=\mathbb{G}_{a} gives a simple example) and therefore fail to be representable on Sm_(k)\mathrm{Sm}_{k}. The situation above where A=G_(m)\mathbf{A}=\mathbb{G}_{m} provides the prototypical example of a sheaf whose cohomology is A^(1)\mathbb{A}^{1}-invariant (here the zeroth cohomology is the presheaf of units, which is even A^(1)\mathbb{A}^{1}-invariant on reduced schemes). Following Morel and Voevodsky, we distinguish the cases where sheaf cohomology is A^(1)\mathbb{A}^{1}-invariant.
Definition 3.9. A sheaf of groups G\mathbf{G} on Sm_(k)\operatorname{Sm}_{k} is called strongly A^(1)\mathbb{A}^{1}-invariant if for i=0,1i=0,1 the functors H_(Nis)^(i)(-,G)\mathrm{H}_{\mathrm{Nis}}^{i}(-, \mathbf{G}) on Sm_(k)\mathrm{Sm}_{k} are A^(1)\mathbb{A}^{1}-invariant. A sheaf of abelian groups A\mathbf{A} on Sm_(k)\mathrm{Sm}_{k} is called strictly A^(1)\mathbb{A}^{1}-invariant if for all i >= 0i \geq 0 the functors H_(Nis)^(i)(-,A)\mathrm{H}_{\mathrm{Nis}}^{i}(-, \mathbf{A}) on Sm_(k)\mathrm{Sm}_{k} are A^(1)\mathbb{A}^{1}-invariant.
The fundamental work of Morel, which we will review shortly, demonstrates the key role played by strongly and strictly A^(1)\mathbb{A}^{1}-invariant sheaves. Nevertheless, various natural functors of geometric origin fail to be A^(1)\mathbb{A}^{1}-invariant on smooth schemes.
Example 3.10. If r >= 2r \geq 2, then the functor H_(Nis)^(1)(-,GL_(r))\mathrm{H}_{\mathrm{Nis}}^{1}\left(-, \mathrm{GL}_{r}\right) fails to be A^(1)\mathbb{A}^{1}-invariant on all schemes. For an explicit example, consider the simplest case. By a theorem of DedekindWeber frequently attributed to Grothendieck every rank nn vector bundle on P^(1)\mathbb{P}^{1} is isomorphic to a unique line bundle of the form bigoplus_(i=1)^(n)O(a_(i))\bigoplus_{i=1}^{n} \mathcal{O}\left(a_{i}\right) with the a_(i)a_{i} weakly increasing. On the other hand, consider P^(1)xxA^(1)\mathbb{P}^{1} \times \mathbb{A}^{1} with coordinates tt and xx. The matrix
([t,x],[0,t^(-1)])\left(\begin{array}{cc}
t & x \\
0 & t^{-1}
\end{array}\right)
determines a rank 2 vector bundle on P^(1)xxA^(1)\mathbb{P}^{1} \times \mathbb{A}^{1} whose restriction to P^(1)xx0\mathbb{P}^{1} \times 0 is O(1)o+O(-1)\mathcal{O}(1) \oplus \mathcal{O}(-1) and whose restriction to P^(1)xx1\mathbb{P}^{1} \times 1 is Oo+O\mathcal{O} \oplus \mathcal{O}. In contrast, Lindel's theorem affirming the BassQuillen conjecture in the geometric case shows that H_(Nis)^(1)(-,GL_(r))\mathrm{H}_{\mathrm{Nis}}^{1}\left(-, \mathrm{GL}_{r}\right) is A^(1)\mathbb{A}^{1}-invariant on affines. The next result generalizes this last observation.
Theorem 3.11 (Morel, Schlichting, Asok-Hoyois-Wendt). If XX is a smooth affine kk-scheme, then for any r inNr \in \mathbb{N} there are functorial bijections of the form
Remark 3.12. The above result was first established by F. Morel in [40] for r!=2r \neq 2 and kk an infinite, perfect field, and his proof was partly simplified by M. Schlichting whose argument also established the case r=2r=2 [51]. The version above is stated in [13].
In [14,15], it is shown that if G\mathrm{G} is an isotropic reductive group scheme (see [14, DEFINITION 3.3.5] for the definition), then the functor assigning to X inSm_(k)^("aff ")X \in \operatorname{Sm}_{k}^{\text {aff }} the set H_(Nis)^(1)(X,G)\mathrm{H}_{\mathrm{Nis}}^{1}(X, \mathrm{G}) is representable by BG. This observation has a number of consequences, e.g., the following result about quadrics (see Examples 3.7 and 3.8).
Theorem 3.14 ([1,14,15]). For any integer i >= 1i \geq 1 and any X inSm_(k)^("aff ")X \in \mathrm{Sm}_{k}^{\text {aff }}, the comparison map
3.5. Postnikov towers, connectedness and strictly A^(1)\mathbb{A}^{1}-invariant sheaves
Recall from Definition 3.9 the notion of strongly or strictly A^(1)\mathbb{A}^{1}-invariant sheaves of groups. F. Morel showed that such sheaves can be thought of as "building blocks" for the unstable A^(1)\mathbb{A}^{1}-homotopy category. Morel's foundational works [39,40][39,40] can be viewed as a careful analysis of strictly and strongly A^(1)\mathbb{A}^{1}-invariant sheaves of groups and the relationship between the two notions. More precisely, Morel showed that working over a perfect
field kk, the A^(1)\mathbb{A}^{1}-homotopy sheaves of a motivic space are always strongly A^(1)\mathbb{A}^{1}-invariant, and that strongly A^(1)\mathbb{A}^{1}-invariant sheaves of abelian groups are automatically strictly A^(1)\mathbb{A}^{1}-invariant.
Example 3.15. Some examples of A^(1)\mathbb{A}^{1}-invariant sheaves that will appear in the sequel are:
unramified Milnor K-theory sheaves K_(i)^(M),i >= 0\mathbf{K}_{i}^{M}, i \geq 0 (see [50, coRoLLARY 6.5, PRoPOSItION 8.6] where, more generally, it is shown that any Rost cycle module gives rise to a strictly A^(1)\mathbb{A}^{1}-invariant sheaf);
the Witt sheaf W\mathbf{W} or unramified powers of the fundamental ideal in the Witt ring I^(j)\mathbf{I}^{j}, j >= 0j \geq 0 (this follows from [46]); and
unramified Milnor-Witt K-theory sheaves K_(i)^(MW),i inZ\mathbf{K}_{i}^{M W}, i \in \mathbb{Z} (see [40, cHAPTER 3] for this assertion, or [31, COROLLARY 8.5,PROPOSITION 9.1] where this observation is generalized to so-called Milnor-Witt cycle modules).
3.16 (Moore-Postnikov factorizations). There is an analog of the Moore-Postnikov factorization of a map f:ErarrBf: \mathscr{E} \rightarrow \mathscr{B} of spaces along the lines described in Section 2. For concreteness we discuss the case where E\mathscr{E} and B\mathscr{B} are A^(1)\mathbb{A}^{1}-connected and ff induces an isomorphism on A^(1)\mathbb{A}^{1}-fundamental sheaves of groups for some choice of base-point in E\mathscr{E}.
Given ff as above, there are tau_( <= i)f inSpc_(k)\tau_{\leq i} f \in \operatorname{Spc}_{k} together with maps Erarrtau_( <= i)f,tau_( <= i)f rarrB\mathscr{E} \rightarrow \tau_{\leq i} f, \tau_{\leq i} f \rightarrow \mathscr{B} and tau_( <= i)f rarrtau_( <= i-1)f\tau_{\leq i} f \rightarrow \tau_{\leq i-1} f fitting into a diagram of exactly the same form as (2.1) (replacing EE by E\mathscr{E} and BB by B)\mathscr{B}). The relevant properties of this presentation are similar to those sketched before (replacing homotopy groups by homotopy sheaves), together with a homotopy pullback diagram of exactly the same form as (2.2). We refer to this tower as the A^(1)\mathbb{A}^{1}-Moore-Postnikov tower of ff and the reader may consult [40, APPENDIX B] or [5, §6] for a more detailed presentation.
If XX is a smooth scheme, then a map psi:X rarrB\psi: X \rightarrow \mathscr{B} lifts to tilde(psi):X rarrE\tilde{\psi}: X \rightarrow \mathscr{E} if and only if lifts exist at each stage of the tower, i.e., if and only if a suitable obstruction vanishes. These obstructions are, by construction, valued in Nisnevich cohomology on XX with values in a strictly A^(1)\mathbb{A}^{1}-invariant sheaf (see [5,§6][5, \S 6] for a more detailed explanation).
By analogy with the situation in topology, we will use the A^(1)\mathbb{A}^{1}-Moore-Postnikov factorization to study lifting problems by means of obstruction theory. The relevant obstructions will lie in cohomology groups of a smooth scheme with coefficients in a strictly A^(1)\mathbb{A}^{1}-invariant sheaf. This motivates the following definition.
Definition 3.17. Let XX be a smooth kk-scheme. We say that XX has A^(1)\mathbb{A}^{1}-cohomological dimension <= d\leq d if for any integer i > di>d and any strictly A^(1)\mathbb{A}^{1}-invariant sheaf F,H_(Nis)^(i)(X,F)=0\mathbf{F}, \mathrm{H}_{\mathrm{Nis}}^{i}(X, \mathbf{F})=0. In that case, we write cd_(A^(1))(X) <= dc d_{\mathbb{A}^{1}}(X) \leq d.
Example 3.18. If XX is a smooth kk-scheme of dimension dd, then XX necessarily has A^(1)\mathbb{A}^{1}-cohomological dimension <= d\leq d as well. Since A^(n)\mathbb{A}^{n} has A^(1)\mathbb{A}^{1}-cohomological dimension <= 0\leq 0, the A^(1)\mathbb{A}^{1}-cohomological dimension can be strictly smaller than Krull dimension; Example 3.6 gives numerous other such examples.
3.6. Complex realization
Assume kk is a field that admits an embedding iotaC:k↪C\iota \mathbb{C}: k \hookrightarrow \mathbb{C}. The functor that assigns to a smooth kk-variety XX the complex manifold X(C)X(\mathbb{C}) equipped with its classical topology extends to a complex realization functor
where H\mathrm{H} is the usual homotopy category of topological spaces [41, §3.3]. By construction, complex realization preserves finite products and homotopy colimits. It follows that the complex realization of the motivic sphere S^(p,q)S^{p, q} is the ordinary sphere S^(p+q)S^{p+q}, and consequently the complex realization functor induces group homomorphisms of the form
Suppose XX is any kk-scheme admitting a complex embedding and fix such an embedding. Write Vect_(r)^("top ")(X)\operatorname{Vect}_{r}^{\text {top }}(X) for the set of isomorphism classes of complex topological vector bundles on XX. There is a function
sending an algebraic vector bundle EE over XX to the topological vector bundle on X(C)X(\mathbb{C}) attached to the base change of EE to X_(C)X_{\mathbb{C}}. We will say that an algebraic vector bundle is algebraizable if it lies in the image of this map.
As rank rr topological vector bundles are classified by the set [X(C),BU(r)][X(\mathbb{C}), \mathrm{B} U(r)] of homotopy classes of maps from X(C)X(\mathbb{C}) to the complex Grassmannian, it follows that the function of the preceding paragraph factors as
Theorem 3.11 implies that the first map is a bijection if XX is a smooth affine kk-scheme (or, alternatively, if r=1r=1 ). More generally, combining Theorem 3.11 and Lemma 3.3 one knows that any element of [X,Gr_(r)]_(A^(1))\left[X, \mathrm{Gr}_{r}\right]_{\mathbb{A}^{1}} may be represented by an actual rank rr vector bundle on any Jouanolou device tilde(X)\tilde{X} of XX; this suggests the following definition.
Definition 3.19. If XX is a smooth kk-scheme, then by a rank rr motivic vector bundle on XX we mean an element of the set [X,Gr_(r)]_(A^(1))\left[X, \mathrm{Gr}_{r}\right]_{\mathbb{A}^{1}}.
Question 3.20. If XX is a smooth complex algebraic variety, then which topological vector bundles are algebraizable (resp. motivic)?
4. OBSTRUCTION THEORY AND VECTOR BUNDLES
In order to apply the obstruction theory described in the previous sections to analyze algebraic vector bundles, we need more information about the structure of the classifying space BGL_(n)\mathrm{BGL}_{n} including information about its A^(1)\mathbb{A}^{1}-homotopy sheaves, and the structure of the homotopy fiber of the stabilization map BGL_(n)rarrBGL_(n+1)\mathrm{BGL}_{n} \rightarrow \mathrm{BGL}_{n+1} induced by the map GL_(n)rarrGL_(n+1)\mathrm{GL}_{n} \rightarrow \mathrm{GL}_{n+1} sending an invertible matrix XX to the block matrix diag(1,X)\operatorname{diag}(1, X).
4.1. The homotopy sheaves of the classifying space of BGL_(n)\mathrm{BGL}_{\boldsymbol{n}}
We observed earlier that BGL_(1)=BG_(m)\mathrm{BGL}_{1}=\mathrm{B} \mathbb{G}_{m} is an Eilenberg-Mac Lane space for the sheaf G_(m)\mathbb{G}_{m} : it is A^(1)\mathbb{A}^{1}-connected, and has exactly 1 nonvanishing A^(1)\mathbb{A}^{1}-homotopy sheaf in degree 1 , which is isomorphic to G_(m)\mathbb{G}_{m}. For n >= 1n \geq 1, the analysis of homotopy sheaves of BGL_(n)\mathrm{BGL}_{n} uses several ingredients. First, Morel-Voevodsky observed that BGL=colim_(n)BGL_(n)\mathrm{BGL}=\operatorname{colim}_{n} \mathrm{BGL}_{n} (for the inclusions described above) represents (reduced) algebraic K-theory after [41, 84 THEOREM 3.13]. Second, Morel observed that there is an A^(1)\mathbb{A}^{1}-fiber sequence of the form
and that A^(n+1)\\0\mathbb{A}^{n+1} \backslash 0 is A^(1)-(n-1)\mathbb{A}^{1}-(n-1)-connected. Furthermore, Morel computed [40] the first nonvanishing A^(1)\mathbb{A}^{1}-homotopy sheaf of A^(n+1)\\0\mathbb{A}^{n+1} \backslash 0 in terms of what he called Milnor-Witt K-theory sheaves (Example 3.15).
Putting these ingredients together, one deduces
pi_(i)^(A^(1))(BGL_(n))~=K_(i)^(Q),quad1 <= i <= n-1\pi_{i}^{\mathbb{A}^{1}}\left(\mathrm{BGL}_{n}\right) \cong \mathbf{K}_{i}^{Q}, \quad 1 \leq i \leq n-1
where K_(i)^(Q)\mathbf{K}_{i}^{Q} is the (Nisnevich) sheafification of the Quillen K-theory presheaf on Sm_(k)\mathrm{Sm}_{k}. Following terminology from topology, sheaves in this range are called stable, and the case i=ni=n is called the first unstable homotopy sheaf. In [3], we described the first unstable homotopy sheaf of BGL_(n)\mathrm{BGL}_{n}.
The group scheme GL_(n)\mathrm{GL}_{n} maps to GL_(n)(C)\mathrm{GL}_{n}(\mathbb{C}) under complex realization; the latter is homotopy equivalent to U(n)U(n). For context, we recall some facts about homotopy of U(n)U(n). A classical result of Bott, refining results of Borel-Hirzeburch [24, THEOREM 25.8] asserts that the image of pi_(2n)(BU(n))\pi_{2 n}(B U(n)) in H_(2n)(BU(n))H_{2 n}(B U(n)) is divisible by precisely (n-1)(n-1) ! [25]. This result implies the assertion that pi_(2n)(U(n))=n\pi_{2 n}(U(n))=n !.
Complex realization yields a map pi_(n,n)^(A^(1))(GL_(n))rarrpi_(2n)(U(n))\pi_{n, n}^{\mathbb{A}^{1}}\left(\mathrm{GL}_{n}\right) \rightarrow \pi_{2 n}(U(n)). One can view the celebrated "Suslin matrices" [59] as providing an algebro-geometric realization of the generator of pi_(2n)(U(n))\pi_{2 n}(U(n)). Analyzing the fiber sequence of (4.1) and putting all of the ingredients above together, we obtain the following result (we refer the reader to Example 3.15 for notation).
Theorem 4.1 ([3, THEOREM 1.1]). Assume kk is a field that has characteristic not equal to 2. For any integer n >= 2n \geq 2, there are strictly A^(1)\mathbb{A}^{1}-invariant sheaves S_(n)\mathbf{S}_{n} fitting into exact sequences of the form:
(1) there is a canonical epimorphism K_(n)^(M)//(n-1)!rarrS_(n)\mathbf{K}_{n}^{M} /(n-1)!\rightarrow \mathbf{S}_{n} which becomes an isomorphism after n-2n-2 contractions (see [3,§2.3][3, \S \mathbf{2} . \mathbf{3}] for this terminology);
(2) there is a canonical epimorphism S_(n)rarrK_(n)^(M)//2\mathbf{S}_{n} \rightarrow \mathbf{K}_{n}^{M} / 2 such that the composite
(3) the fiber product is taken over the epimorphism S_(n+1)rarrK_(n+1)^(M)//2\mathbf{S}_{n+1} \rightarrow \mathbf{K}_{n+1}^{M} / 2 and a sheafified version of Milnor's homomorphism I^(n+1)rarrK_(n+1)^(M)//2\mathbf{I}^{n+1} \rightarrow \mathbf{K}_{n+1}^{M} / 2.
Moreover, if kk admits a complex embedding, then the map
Bott's refinement of the theorem of Borel-Hirzebruch turns out to have an algebrogeometric interpretation. Indeed, in joint work with T. B. Williams [12] we showed that S_(n)\mathbf{S}_{n} can described using a "Hurewicz map" analyzed by Andrei Suslin [60]. Suslin's conjecture on the image of this map is equivalent to the following conjecture.
Conjecture 4.2 (Suslin's factorial conjecture). The canonical epimorphism K_(n)^(M)\mathbf{K}_{n}^{M} / (n-1)!rarrS_(n)(n-1)!\rightarrow \mathbf{S}_{n} is an isomorphism.
Remark 4.3. The conjecture holds tautologically for n=2n=2. For n=3n=3, Suslin observed the conjecture was equivalent to the Milnor conjecture on quadratic forms, which was resolved later independently by Merkurjev-Suslin and Rost. The conjecture was established for n=5n=5 in "most" cases in [12] (see the latter for a precise statement); this work relies heavily on the computation by Østvær-Röndigs-Spitzweck of the motivic stable 1-stem [49].
4.2. Splitting bundles, Euler classes, and cohomotopy
Morel's computations around A^(n)\\0\mathbb{A}^{n} \backslash 0 in conjunction with the fiber sequences of (4.1) allow a significant improvement of Serre's celebrated splitting theorem for smooth affine varieties over a field that we stated in the introduction.
Proof of the motivic Serre Splitting Theorem 1.1. Suppose XX is a smooth affine kk-variety having A^(1)\mathbb{A}^{1}-cohomological dimension <= d\leq d, and suppose xi:X rarrBGL_(r)\xi: X \rightarrow \mathrm{BGL}_{r} classifies a rank r > dr>d vector bundle on XX. We proceed by analyzing the A^(1)\mathbb{A}^{1}-Moore-Postnikov factorization of the stabilization map (4.1) with n=r-1n=r-1. In that case, combining the fact that A^(r)\\0\mathbb{A}^{r} \backslash 0 is A^(1)\mathbb{A}^{1} - (r-2)(r-2)-connected and the A^(1)\mathbb{A}^{1}-cohomological dimension assumption on XX, one sees all obstructions to splitting vanish.
Remark 4.4. The proof of this result does not rely on the Serre splitting theorem. Since A^(1)\mathbb{A}^{1}-cohomological dimension can be strictly smaller than Krull dimension (Example 3.18), this statement is strictly stronger than Serre splitting. Importantly, the improvement achieved here seems inaccessible to classical techniques.
The algebro-geometric splitting problem in corank 0 on smooth affine varieties of dimension dd over a field kk has been analyzed by many authors. When kk is an algebraically closed field, M. P. Murthy proved that the top Chern class in Chow groups is the only obstruction to splitting [42]. When kk is not algebraically closed, vanishing of the top Chern class is known to be insufficient to guarantee splitting, and Nori proposed some ideas to analyze this situation. His ideas led Bhatwadekar and Sridharan [23] to introduce what they called Euler class groups and to provide one explicit "generators and relations" answer to this question. At the same time, F. Morel proposed an approach to the splitting problem in corank 0 , which we recall here.
Theorem 4.5 (Morel's splitting theorem [40, THEOREM 1.32]). Assume kk is a field and XX is a smooth affine kk-variety of A^(1)\mathbb{A}^{1}-cohomological dimension <= d\leq d. If E\mathcal{E} is a rank dd vector bundle on XX, then E\mathcal{E} splits off a free rank 1 summand if and only if an Euler class e(E)inH_(Nis)^(d)(X,K_(d)^(MW)(det E))e(\mathcal{E}) \in \mathrm{H}_{\mathrm{Nis}}^{d}\left(X, \mathbf{K}_{d}^{M W}(\operatorname{det} \mathcal{E})\right) vanishes.
Remark 4.6. The Euler class of Theorem 4.5 is precisely the first nonvanishing obstruction class, as described in Paragraph 3.16. A related "cohomological" approach to the splitting problem in corank 0 was proposed by Barge-Morel [20] and analyzed in the thesis of the second author [29]. The cohomological approach was in most cases shown to be equivalent to the "obstruction-theoretic" approach in [6]. We also refer the reader to [51] for related results on the theory of Euler classes, extending also to singular varieties.
The next result shows that the relationship between Euler classes à la BhatwadekarSridharan and Euler classes à la Morel is mediated by another topologically inspired notion: cohomotopy (at least for bundles of trivial determinant).
Theorem 4.7 ([8, Ñ‚heorem 1]). Suppose kk is a field, nn and dd are integers, n >= 2n \geq 2, and XX is a smooth affine kk-scheme of dimension d <= 2n-2d \leq 2 n-2. Write E^(n)(X)\mathrm{E}^{n}(X) for the BhatwadekarSridharan Euler class group.
The set [X,Q_(2n)]_(A^(1))\left[X, Q_{2 n}\right]_{\mathbb{A}^{1}} carries a functorial abelian group structure;
where the "Segre class" homomorphism s is surjective and an isomorphism if kk is infinite and d >= 2d \geq 2, and the Hurewicz homomorphism hh is an isomorphism if d <= nd \leq n.
Remark 4.8. The group structure on [X,Q_(2n)]_(A^(1))\left[X, Q_{2 n}\right]_{\mathbb{A}^{1}} is an algebro-geometric variant of Borsuk's group structure on cohomotopy. The second point of the statement includes the algebrogeometric analog of the Hopf classification theorem from topology.
4.3. The next nontrivial A^(1)\mathbb{A}^{1}-homotopy sheaf of spheres
In Section 2.2 we described a cohomological approach to the splitting problem in corank 1 for smooth closed manifolds of dimension dd; this approach relied on the computa-
tion of pi_(d)(S^(d-1))\pi_{d}\left(S^{d-1}\right). In order to analyze the algebro-geometric splitting problem in corank 1 using the A^(1)\mathbb{A}^{1}-Moore-Postnikov factorization we will need as input further information about the homotopy sheaves of A^(d)\\0\mathbb{A}^{d} \backslash 0. We now describe known results in this direction. For technical reasons, we assume 2 is invertible in what follows.
4.3.1. The KOK O-degree map
In classical algebraic topology, all of the "low degree" elements in the homotopy of spheres can be realized by constructions of "linear algebraic" nature. The situation in algebraic geometry appears to be broadly similar. The first contribution to the "next" nontrivial homotopy sheaves of motivic spheres requires recalling the geometric formulation of Bott periodicity for Hermitian K-theory given by Schlichting-Tripathi.
We write O\mathrm{O} for the infinite orthogonal group. In topology, Bott periodicity identifies the 8 -fold loop space of O\mathrm{O} with itself and identifies the intermediate loop spaces in concrete geometric terms. In algebraic geometry, Schlichting and Tripathi proved that the 4-fold P^(1)\mathbb{P}^{1}-loop space Omega_(P^(1))^(4)O\Omega_{\mathbb{P}^{1}}^{4} \mathrm{O} also coincides with O\mathrm{O} and realized suitable intermediate loop spaces: Omega_(P^(1))^(n)O\Omega_{\mathbb{P}^{1}}^{n} \mathrm{O} is isomorphic to GL / O when n=1,Spn=1, \mathrm{Sp} when n=2n=2 and GL //Sp/ \mathrm{Sp} when n=3n=3, where Sp\mathrm{Sp} is the stable sympletic group, GL/O is the ind-variety of invertible symmetric matrices, and GL/Sp is the ind-variety of invertible skew-symmetric matrices [52,
THEOREMS 8.2 AND 8.4].
A slight modification of the Suslin matrix construction [59, LEMMA 5.3] yields a map
u_(n):Q_(2n-1)rarrOmega_(P^(1))^(-n)Ou_{n}: Q_{2 n-1} \rightarrow \Omega_{\mathbb{P}^{1}}^{-n} O
called the (unstable) KO-degree map in weight nn that was analyzed in detail in [7]. The terminology stems from the fact that this map stabilizes to the "unit map from the sphere spectrum to the Hermitian K-theory spectrum" in an appropriate sense. The scheme Q_(2n-1)Q_{2 n-1} is A^(1)\mathbb{A}^{1} - (n-2)(n-2)-connected by combining the weak equivalence of Example 3.7 and Morel's connectivity results for A^(n)\\0\mathbb{A}^{n} \backslash 0. Thus, u_(n)u_{n} factors through the A^(1)-(n-2)\mathbb{A}^{1}-(n-2)-connected cover of Omega_(P^(1))^(-n)O\Omega_{\mathbb{P}^{1}}^{-n} O.
Taking homotopy sheaves on both sides, there are induced morphisms
This homomorphism is trivial if i < n-1i<n-1 by connectivity estimates. If i=n-1i=n-1, via Morel's calculations one obtains a morphism K_(n)^(MW)rarrGW_(n)^(n)\mathbf{K}_{n}^{\mathrm{MW}} \rightarrow \mathbf{G W}_{n}^{n} whose sections over finitely generated field extensions of kk can be viewed as a quadratic enhancement of the "natural" map from Milnor KK-theory to Quillen KK-theory defined by symbols; we will refer to it as the natural homomorphism (the natural homomorphism is known to be an isomorphism if n <= 4n \leq 4; the case n <= 2n \leq 2 is essentially Suslin's, n=3n=3 is [7, THEOREM 4.3.1], and n=4n=4 is unpublished work of O. Röndigs).
where the right-hand term is by definition a higher Grothendieck-Witt sheaf (obtained by sheafifying the corresponding higher Grothendieck-Witt presheaf on Sm_(k)\mathrm{Sm}_{k} ). The above map
is an epimorphism for n=2,3n=2,3 and it follows from these observations that the morphism is an epimorphism after (n-3)(n-3) contractions [7, THEOREM 4.4.5].
4.3.2. The motivic J-homomorphism
The classical J-homomorphism has an algebro-geometric counterpart that yields the second contribution to the "next" homotopy sheaf of motivic spheres. The standard action of SL_(n)\mathrm{SL}_{n} on A^(n)\mathbb{A}^{n} extends to an action on the one-point compactification P^(n)//P^(n-1)\mathbb{P}^{n} / \mathbb{P}^{n-1}. The latter space is a motivic sphere P^(1^(^^n))\mathbb{P}^{1^{\wedge n}} and thus one obtains a map
Sigma_(P^(1))^(n)SL_(n)rarrP^(1^(^^n))\Sigma_{\mathbb{P}^{1}}^{n} S L_{n} \rightarrow \mathbb{P}^{1^{\wedge n}}
As SL_(n)\mathrm{SL}_{n} is A^(1)\mathbb{A}^{1}-connected, it follows that Sigma_(P^(1))^(n)SL_(n)\Sigma_{\mathbb{P}^{1}}^{n} \mathrm{SL}_{n} is A^(1)\mathbb{A}^{1} - nn-connected.
The first nonvanishing A^(1)\mathbb{A}^{1}-homotopy sheaf appears in degree n+1n+1; for n=2n=2, it is isomorphic to K_(4)^(MW)\mathbf{K}_{4}^{M W}, while for n >= 3n \geq 3 it is isomorphic to K_(n+2)^(M)\mathbf{K}_{n+2}^{M}; this follows from A^(1)\mathbb{A}^{1}-Hurewicz theorem combined with [17, PROPOSItIoN 3.3.9] using the fact that pi_(1)^(A^(1))(SL_(n))=K_(2)^(M)\pi_{1}^{\mathbb{A}^{1}}\left(\mathrm{SL}_{n}\right)=\mathbf{K}_{2}^{M} for n >= 3n \geq 3 and properties of the A^(1)\mathbb{A}^{1}-tensor product [17, LEMMA 5.1.8].
Combining the above discussion with that of the previous section, we see that for n >= 3n \geq 3, we may consider the composite maps K_(n+2)^(M)rarrpi_(n+1)^(A^(1))(P^(1^^n))rarrGW_(n+1)^(n)\mathbf{K}_{n+2}^{M} \rightarrow \pi_{n+1}^{\mathbb{A}^{1}}\left(\mathbb{P}^{1 \wedge n}\right) \rightarrow \mathbf{G W}_{n+1}^{n}; this composite is known to be zero, but the map induced by the J-homomorphism fails to be injective. Instead, it factors through a morphism
Furthermore, the map on the right fails to be surjective. The unstable description above is not present in the literature, but it is equivalent to the results stated in [12]. In [49], the stable motivic 1-stem was computed in the terms above: the above sequence is exact on the left stably. The next result compares the unstable group to the corresponding stable group.
Theorem 4.9. For any integer n >= 3n \geq 3, the kernel U_(n+1)\mathbf{U}_{n+1} of the stabilization map
is a direct summand; the stabilization map is an isomorphism if n=3n=3, i.e., U_(4)=0\mathbf{U}_{4}=0.
Conjecture 4.10. For n >= 4n \geq 4, the sheaf U_(n+1)\mathbf{U}_{n+1} is zero.
Remark 4.11. Conjecture 4.10 would follow from a suitable version of the Freudenthal suspension theorem for P^(1)\mathbb{P}^{1}-suspension.
4.4. Splitting in corank 1
Using the results above, we can analyze the splitting problem for vector bundles in corank 1. The expected result was posed as a question by Murthy [43, P. 173] which we stated in the introduction as Conjecture 1.2. Murthy's conjecture is trivial if d=2d=2. In [4] and [5] we established the following result, which reduces Murthy's question to Conjecture 4.10.
Theorem 4.12. Let XX be a smooth affine scheme of dimension d >= 2d \geq 2 over an algebraically closed field kk. A rank d-1d-1 vector bundle &\& on XX splits off a trivial rank 1 summand if and
only if c_(d-1)(E)inCH^(d-1)(X)c_{d-1}(\mathcal{E}) \in \mathrm{CH}^{d-1}(X) is trivial and a secondary obstruction
vanishes. This secondary obstruction vanishes if d=3,4d=3,4 or if Conjecture 4.10 has a positive answer.
To establish this result, one uses the assumptions that XX is smooth affine of Krull dimension dd and kk is algebraically closed in a strong way. Indeed, these assertions can be leveraged to show that the primary obstruction, which is a priori an Euler class, actually coincides with the (d-1)(d-1) st Chern class. The secondary obstruction can be described by Theorem 4.9 and the form of the secondary obstruction is extremely similar to Liao's description in Section 2.2: it is a coset in Ch^(d)(X)//(Sq^(2)+c_(1)(E)uu)Ch^(d-1)(X)\mathrm{Ch}^{d}(X) /\left(\mathrm{Sq}^{2}+c_{1}(\mathcal{E}) \cup\right) \mathrm{Ch}^{d-1}(X) where Ch^(i)(X)=CH^(i)(X)//2\mathrm{Ch}^{i}(X)=\mathrm{CH}^{i}(X) / 2. Once more, the assumptions on XX guarantee that Ch^(d)(X)\mathrm{Ch}^{d}(X) is trivial and thus the secondary obstruction is so as well.
4.5. The enumeration problem
If a vector bundle EE splits off a free rank 1 summand, then another natural question is to enumerate the possible E^(')E^{\prime} that become isomorphic to EE after adding a free rank 1 summand. This problem may also be analyzed in homotopy theoretic terms as it amounts to enumerating the number of distinct lifts. This kind of problem was studied in detail in topology by James and Thomas [33] and the same kind of analysis can be pursued in algebraic geometry.
The history of the enumeration problem in algebraic geometry goes back to early days of algebraic K-theory. Indeed, the Bass-Schanuel cancellation theorem [22] solves the enumeration problem for bundles of negative corank. Suslin's celebrated cancellation theorem [58] solved the enumeration problem in corank 0 . In all of these statements, "cancellation" means that there is a unique lift. On the other hand, Mohan Kumar observed [38] that for bundles of corank 2 , uniqueness was no longer true in general. Nevertheless, Suslin conjectured that the enumeration problem had a particularly nice solution in corank 1.
Conjecture 4.13 (Suslin's cancellation conjecture). If kk is an algebraically closed field, and XX is a smooth affine kk-scheme of dimension d >= 2d \geq 2. If E\mathcal{E} and E^(')\mathcal{E}^{\prime} are corank 1 bundles that become isomorphic after addition of a trivial rank 1 summand, then E\mathcal{E} and E^(')\mathcal{E}^{\prime} are isomorphic.
The above conjecture is trivial when d=2d=2. It was established for E\mathcal{E} the trivial bundle of rank d-1d-1 in [30] and d=dim(X)d=\operatorname{dim}(X) under the condition that (d-1)(d-1) ! is invertible in kk. The above conjecture was also established for d=3d=3 in [4] (assuming 2 is invertible in kk ). Paralleling the results of James-Thomas in topology [33], P. Du was able to prove in [27] that Suslin's question has a positive answer for oriented vector bundles in case the cohomology group H_(Nis)^(d)(X,pi_(d)^(A^(1))(A^(d)\\0))\mathrm{H}_{\mathrm{Nis}}^{d}\left(X, \pi_{d}^{\mathbb{A}^{1}}\left(\mathbb{A}^{d} \backslash 0\right)\right) vanishes. This vanishing statement would follow immediately from Conjecture 4.10.
5. VECTOR BUNDLES: NONAFFINE VARIETIES AND ALGEBRAIZABILITY
In this final section, we survey some joint work with M. J. Hopkins related to the classification of motivic vector bundles (see Definition 3.19), its relationship to the algebraizability question (see Question 3.20), and investigate the extent to which A^(1)\mathbb{A}^{1}-homotopy theory can be used to analyze vector bundles on projective varieties.
5.1. Descent along a Jouanolou device
If XX is a smooth algebraic kk-variety, then there is always the map
from rank rr vector bundles to rank rr motivic vector bundles. When XX is affine, Theorem 3.11 guarantees that this map is a bijection, and examples show that the map fails to be an isomorphism outside of this case. Nevertheless, it is very interesting to try to quantify the failure of the above map to be a bijection.
If pi: tilde(X)rarr X\pi: \tilde{X} \rightarrow X is a Jouanolou device for XX, then it follows from the definitions that the map (5.1) coincides with pi^(**):Vect_(r)(X)rarrVect_(r)( tilde(X))\pi^{*}: \operatorname{Vect}_{r}(X) \rightarrow \operatorname{Vect}_{r}(\tilde{X}) under the bijection of Theorem 3.11. The morphism pi\pi is faithfully flat by construction, and therefore, vector bundles on XX are precisely vector bundles on tilde(X)\tilde{X} equipped with a descent datum along pi\pi.
Since pi: tilde(X)rarr X\pi: \tilde{X} \rightarrow X is an affine morphism, it follows that tilde(X)xx_(X) tilde(X)\tilde{X} \times_{X} \tilde{X} is itself an affine scheme, and the two projections p_(1),p_(2): tilde(X)xx_(X) tilde(X)rarr tilde(X)p_{1}, p_{2}: \tilde{X} \times_{X} \tilde{X} \rightarrow \tilde{X} are A^(1)\mathbb{A}^{1}-weak equivalences. Thus, pullbacks p_(1)^(**)p_{1}^{*} and p_(2)^(**)p_{2}^{*} are bijections on sets of isomorphism classes of vector bundles. In fact, since the relative diagonal map splits the two projections, the two pullbacks actually coincide on isomorphism classes. In descent-theoretic terms, these observations mean that any vector bundle E\mathscr{E} on tilde(X)\tilde{X} can always be equipped with an isomorphism p_(1)^(**)Erarr"∼"p_(2)^(**)Ep_{1}^{*} \mathscr{E} \xrightarrow{\sim} p_{2}^{*} \mathscr{E}, i.e., a predescent datum. Thus, the only obstruction to descending a vector bundle along pi\pi is whether one may choose a predescent datum that actually satisfies the cocycle condition. With this observation in mind, it seems natural to analyze the question of whether every vector bundle can be equipped with a descent datum along pi\pi.
Question 5.1. If XX is a smooth kk-variety and pi: tilde(X)rarr X\pi: \tilde{X} \rightarrow X is a Jouanolou device for XX, then is the pull-back map
Theorem 5.2 (Asok, Fasel, Hopkins). Suppose XX is a smooth projective kk-variety of dimension d. If either (i) d <= 2d \leq 2 or (ii) kk is algebraically closed and d <= 3d \leq 3, then Question 5.1 admits a positive answer, i.e., every vector bundle on tilde(X)\tilde{X} admits a descent datum relative to pi\pi.
5.2. Algebraizability I: obstructions
If XX is a smooth complex algebraic variety, then we considered the map
and posed the question of characterizing its image. We observed that this map factors through the set of motivic vector bundles, so one necessary condition for a topological vector bundle to be algebraizable is that it admits a motivic lift. In particular, this means that the Chern classes of the topological vector bundle in integral cohomology must lie in the image of the cycle class map. It is natural to ask if algebraizability of Chern classes is sufficient to guarantee that a vector bundle admits a motivic left.
In case XX is projective, this question has been for instance studied in [53] where it is proved that any vector bundle with algebraic Chern classes is algebraizable if dim(X)=2\operatorname{dim}(X)=2. In case of projective threefolds, positive results are given by Atiyah-Rees and Bănică-Putinar respectively in [18] and [19]. If XX is affine, the works of Swan-Murthy [44] and MurthyKumar [36] show that the answer to the question is positive if XX is of dimension <= 3\leq 3 as a consequence of the following statement: Given any pair (alpha_(1),alpha_(2))inCH^(1)(X)xxCH^(2)(X)\left(\alpha_{1}, \alpha_{2}\right) \in \mathrm{CH}^{1}(X) \times \mathrm{CH}^{2}(X), there exists a vector bundle E\mathcal{E} on XX with c_(i)(E)=alpha_(i)c_{i}(\mathscr{E})=\alpha_{i}. However, in dimension 4 , additional restrictions on Chern classes arise from the action of the motivic Steenrod algebra.
Theorem 5.3 ([10, THEOREM 2]). If XX is a smooth affine 4-fold, then a pair (c_(1),c_(2))in\left(c_{1}, c_{2}\right) \inCH^(1)(X)xxCH^(2)(X)\mathrm{CH}^{1}(X) \times \mathrm{CH}^{2}(X) are Chern classes of a rank 2 bundle on XX if and only if c_(1),c_(2)c_{1}, c_{2} satisfy the additional condition Sq^(2)(c_(2))+c_(1)c_(2)=0\mathrm{Sq}^{2}\left(c_{2}\right)+c_{1} c_{2}=0, where
is the Steenrod squaring operation, and c_(1)c_(2)c_{1} c_{2} is the reduction modulo 2 of the cup product.
Remark 5.4. This obstruction is sufficient to identify topological vector bundles on a smooth affine fourfold XX having algebraic Chern classes which are not algebraizable [10, coRolLARY 3.1.5]. One example of such an XX is provided by the open complement in P^(1)xxP^(3)\mathbb{P}^{1} \times \mathbb{P}^{3} of a suitable smooth hypersurface ZZ of bidegree (3, 4).
5.3. Algebraizability II: building motivic vector bundles
The notion of a cellular space goes back to the work of Dror Farjoun. By a cellular motivic space, we will mean a space that can be built out of the motivic spheres S^(p,q)S^{p, q} by formation of homotopy colimits. It is straightforward to see inductively that P^(n)\mathbb{P}^{n} is cellular. In the presence of cellularity assumptions, many obstructions to producing a motivic lift of a vector bundle vanish and this motivates the following conjecture.
Conjecture 5.5. If XX is a smooth cellular C\mathbb{C}-variety, then the map
Remark 5.6. The conjecture holds for P^(n)\mathbb{P}^{n} for n <= 3n \leq 3 (this follows, for example, from the results of Schwarzenberger and Atiyah-Rees mentioned above); in these cases, bijectivity holds. For P^(4)\mathbb{P}^{4}, the "surjective" formulation of Conjecture 5.5 is known, but the "bijective" formulation is not.
We now analyze Conjecture 5.5 for a class of "interesting" topological vector bundles on P^(n)\mathbb{P}^{n} introduced by E. Rees and L. Smith. We briefly recall the construction of these topological vector bundles here. By a classical result of Serre [54, PROPOsition 11], we know that if pp is a prime, then the pp-primary component of pi_(4p-3)(S^(3))\pi_{4 p-3}\left(S^{3}\right) is isomorphic to Z//p\mathbb{Z} / p, generated by the composite of a generator alpha_(1)\alpha_{1} of the pp-primary component of pi_(2p)(S^(3))\pi_{2 p}\left(S^{3}\right) and the (2p-3)(2 p-3) rd suspension of itself; we will write alpha_(1)^(2)\alpha_{1}^{2} for this class.
The map P^(n)rarrS^(2n)\mathbb{P}^{n} \rightarrow S^{2 n} that collapses P^(n-1)\mathbb{P}^{n-1} to a point determines a function
Rees established that the class alpha_(1)^(2)\alpha_{1}^{2} determines a nontrivial rank 2 vector bundle xi_(p)in\xi_{p} \in[P^(2p-1),BSU(2)]\left[\mathbb{P}^{2 p-1}, \mathrm{BSU}(2)\right]; we will refer to this bundle as a Rees bundle [48]. By construction, xi_(p)\xi_{p} is a nontrivial rank 2 bundle with trivial Chern classes.
The motivation for Rees' construction originated from results of Grauert-Schneider [32]. If the bundles xi_(p)\xi_{p} were algebraizable, then the fact that they have trivial Chern classes would imply they were necessarily unstable by Barth's results on Chern classes of stable vector bundles [21, CORoLLARY 1 P. 127] (here, stability means slope stability in the sense of Mumford). Grauert and Schneider analyzed unstable rank 2 vector bundles on projective space and they aimed to prove that such vector bundles were necessarily direct sums of line bundles; this assertion is now sometimes known as the Grauert-Schneider conjecture. In view of the Grauert-Schneider conjecture, the bundles xi_(p)\xi_{p} should not be algebraizable. On the other hand, one of the motivations for Conjecture 5.5 is the following result.
Theorem 5.7 ([11, THEOREM 2.2.16]). For every prime number pp, the bundle xi_(p)\xi_{p} lifts to a class in [P^(2p-1),Gr_(2)]_(A^(1))\left[\mathbb{P}^{2 p-1}, \mathrm{Gr}_{2}\right]_{\mathbb{A}^{1}}.
Remark 5.8. This is established by constructing motivic homotopy classes lifting alpha_(1)\alpha_{1} and alpha_(1)^(2)\alpha_{1}^{2}. In our situation, the collapse map takes the form
and the lift must come from an element of [S^(n-1,n),SL_(2)]_(A^(1))\left[S^{n-1, n}, S L_{2}\right]_{\mathbb{A}^{1}}. The class alpha_(1)\alpha_{1} can be lifted using ideas related to those discussed in 4.1 in conjunction with a motivic version of Serre's classical pp-local splitting of compact Lie groups [9, THEOREM 2], the resulting lift has the wrong weight to lift to a group as above. Since the class alpha_(1)^(2)\alpha_{1}^{2} is torsion, we can employ a weightshifting mechanism to fix this issue. In this direction, there are host of other vector bundles that are analogous to the Rees bundles that one might investigate from this point of view, e.g., bundles that can be built out of Toda's unstable alpha\alpha-family [62]. Likewise, even the surjectivity assertion in Conjecture 5.5 is unknown for P^(5)\mathbb{P}^{5}.
5.9 (The Wilson Space Hypothesis). To close, we briefly sketch an approach to the resolution of Conjecture 5.5 building on Mike Hopkins' Wilson Space Hypothesis. The latter asserts that the Voevodsky motive of the P^(1)\mathbb{P}^{1}-infinite loop spaces Omega_(P^(1))^(oo)Sigma_(P^(1))^(n)\Omega_{\mathbb{P}^{1}}^{\infty} \Sigma_{\mathbb{P}^{1}}^{n} MGL arising from algebraic cobordism are pure Tate (the space is "homologically even"); this hypothesis is an algebro-geometric version of a result of Steve Wilson on the infinite loop spaces of the classical cobordism spectrum.
The motivic version of the unstable Adams-Novikov resolution for BGL_(r)\mathrm{BGL}_{r} yields a spectral sequence that, under the cellularity assumption on XX should converge to a (completion of) the set of rank rr motivic vector bundles on XX. The resulting spectral sequence can be compared to its topological counterpart and Wilson Space Hypothesis combined with the cellularity assumption on XX would imply that the two spectral sequences coincide.
ACKNOWLEDGMENTS
The authors would like to acknowledge the profound influence of M. Hopkins and F. Morel on their work. Many of the results presented here would not have been possible without their insights and vision. Both authors would also like to thank all their colleagues, coauthors, and friends for their help and support.
FUNDING
Aravind Asok was partially supported by NSF Awards DMS-1802060 and DMS-2101898.
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ARAVIND ASOK
Department of Mathematics, University of Southern California, 3620 S. Vermont Ave., Los Angeles, CA 90089-2532, USA, asok @usc.edu
Bridgeland stability conditions and wall-crossing have provided answers to many questions in algebraic geometry a priori unrelated to derived categories, including hyperkähler varieties-their rational curves, their birational geometry, their automorphisms, and their moduli spaces-, Brill-Noether questions, Noether-Lefschetz loci, geometry of cubic fourfolds, or higher-rank Donaldson-Thomas theory. Our goal is to answer the question: why? What makes these techniques so effective, and what exactly do they add beyond, for example, classical vector bundle techniques?
The common underlying strategy can be roughly summarized as follows. For each stability condition on a derived category D^(b)(X)\mathrm{D}^{\mathrm{b}}(X) of an algebraic variety XX and each numerical class, moduli spaces of semistable objects in D^(b)(X)\mathrm{D}^{\mathrm{b}}(X) exist as proper algebraic spaces. This formalism includes many previously studied moduli spaces: moduli spaces of Gieseker- or slope-stable sheaves, of stable pairs, or of certain equivalences classes of rational curves in XX. The set of stability conditions on D^(b)(X)\mathrm{D}^{\mathrm{b}}(X) has the structure of a complex manifold; when we vary the stability condition, stability of a given object only changes when we cross the walls of a well-defined wall and chamber structure.
The typical ingredients when approaching a problem with stability conditions are the following:
(large volume) There is a point in the space of stability conditions where stable objects have a "classical" interpretation, e.g. as Gieseker-stable sheaves.
(point of interest) There is a point in the space of stability conditions where stability has strong implications, e.g., vanishing properties, or even there is no semistable object of a given numerical class.
(wall-crossing) It is possible to analyze the finite set of walls between these two points, and how stability changes when crossing each wall.
In general, it is quite clear from the problem which are the points of interest, and the main difficulty consists in analyzing the walls. In the ideal situation, which leads to sharp exact results, these walls can be characterized purely numerically; there are only a few such ideal situations, K3 surfaces being one of them. Otherwise, the study of walls can get quite involved, even though there are now a number of more general results available, e.g., a wallcrossing formula for counting invariants arising from moduli spaces.
We illustrate the case of K3 surfaces, or more generally K3 categories, in Section 3, with applications to hyperkähler varieties, to Brill-Noether theory of curves, and to the geometry of special cubic fourfolds. The study of other surfaces or the higher-dimensional case becomes more technical, and the existence of Bridgeland stability conditions is not yet known in full generality. There are weaker notions of stability, which in the threefold case already lead to striking results. We give an overview of this circle of ideas in Section 4, along with three applications related to curves. We give a brief introduction to stability conditions in Section 2, and pose some questions for future research in Section 5.
Derived categories of coherent sheaves on varieties have been hugely influential in recent years; we refer to [17,19,46,74][17,19,46,74] for an overview of the theory. Moduli spaces of sheaves on K3 surfaces have largely been influenced by [61]; we refer to [63] for an overview of applications of these techniques and to [39] for the higher-dimensional case of hyperkähler manifolds. For the original motivation from physics, we refer to [26,41].
Our survey completely omits the quickly developing theory and applications of stability conditions on Kuznetsov components of Fano threefolds. We also will not touch the rich subject of extra structures on spaces of stability conditions, developed, for example, in the foundational papers [22,23]; we also refer to [72] for a symplectic perspective.
2. STABILITY CONDITIONS ON DERIVED CATEGORIES
Recall slope-stability for sheaves on an integral projective curve CC : we set
mu(E)=(deg E)/(rk E)in(-oo,+oo],quad" visualized by "Z(E)=-deg E+irkE\mu(E)=\frac{\operatorname{deg} E}{\operatorname{rk} E} \in(-\infty,+\infty], \quad \text { visualized by } Z(E)=-\operatorname{deg} E+i \mathrm{rk} E
and call a sheaf slope-semistable if every subsheaf F sub EF \subset E satisfies mu(F) <= mu(F)\mu(F) \leq \mu(F). The set of semistable sheaves of fixed rank and degree is bounded and can be parameterized by a projective moduli space. Moreover, semistable sheaves generate Coh(C)\operatorname{Coh}(C), in the sense that every sheaf EE admits a (unique and functorial) Harder-Narasimhan (HN) filtration
with E_(l)//E_(l-1)E_{l} / E_{l-1} semistable, and mu(E_(1)) > mu(E_(2)//E_(1)) > cdots > mu(E_(m)//E_(m-1))\mu\left(E_{1}\right)>\mu\left(E_{2} / E_{1}\right)>\cdots>\mu\left(E_{m} / E_{m-1}\right).
How to generalize this to a variety XX of dimension n >= 2n \geq 2 ? Given a polarization HH, one can define the slope mu_(H)\mu_{H} using H^(n-1)*ch_(1)(E)H^{n-1} \cdot \operatorname{ch}_{1}(E) as the degree. To distinguish, e.g., the slope of the structure sheaf O_(X)\mathcal{O}_{X} from that of an ideal sheaf I_(x)subO_(X)\mathscr{I}_{x} \subset \mathcal{O}_{X} for x in Xx \in X, we can further refine the notion of slope and use lower-degree terms of the Hilbert polynomial p_(E)(m)=chi(E(mH))p_{E}(m)=\chi(E(m H)) as successive tie breakers; this yields Gieseker stability.
One of the key insights in Bridgeland's notion of stability conditions introduced in [20] is that instead we can, in fact, still use a notion of slope-stability, defined as the quotient of "degree" by "rank." The price we have to pay is to replace Coh(X)\operatorname{Coh}(X) by another abelian subcategory A\mathcal{A} of the bounded derived category D^(b)(X)\mathrm{D}^{\mathrm{b}}(X) of coherent sheaves on XX, and to generalize the notions of "degree" and "rank" (combined into a central charge ZZ as above).
To motivate the definition, consider again slope-stability for a curve CC. First, for phi in(0,1]\phi \in(0,1], let P(phi)sub Coh(C)subD^(b)(C)\mathcal{P}(\phi) \subset \operatorname{Coh}(C) \subset \mathrm{D}^{\mathrm{b}}(C) be the category of slope-semistable sheaves EE with Z(E)inR_( > 0)*e^(i pi phi)Z(E) \in \mathbb{R}_{>0} \cdot e^{i \pi \phi}, i.e., of slope mu(E)=-cot(pi phi)\mu(E)=-\cot (\pi \phi), and let P(phi+n)=P(phi)[n]\mathcal{P}(\phi+n)=\mathcal{P}(\phi)[n] for n inZn \in \mathbb{Z} be the set of semistable objects of phase phi+n\phi+n. Every complex E inD^(b)(C)E \in \mathrm{D}^{\mathrm{b}}(C) has a filtration into its cohomology objects H^(l)(E)[-l]\mathscr{H}^{l}(E)[-l]. We can combine this with the classical HN filtration of H^(l)(E)[-l]\mathscr{H}^{l}(E)[-l] for each ll to obtain a finer filtration for EE where every filtration quotient is semistable, i.e., an object of P(phi)\mathcal{P}(\phi) for phi inR\phi \in \mathbb{R}. The properties of this structure are axiomatized by conditions (1)-(4) in Definition 2.1 below. But crucially it can always be obtained from slope-stability in an abelian category A\mathcal{A}; we just have to generalize the setting AsubD^(b)(A)\mathcal{A} \subset \mathrm{D}^{\mathrm{b}}(\mathcal{A}) to AsubD\mathscr{A} \subset \mathscr{D} being the "heart of a bounded t\mathrm{t}-structure" in a triangulated category.
When combined with the remaining conditions in Definition 2.1, the main payoff are the strong deformation and wall-crossing properties of Bridgeland stability conditions. Given any small deformation of "rank" and "degree" (equivalently, of the central charge ZZ ), we can accordingly adjust the abelian category A\mathcal{A} (or, equivalently, the set of semistable objects P\mathcal{P} ) and obtain a new stability condition. Along such a deformation, moduli spaces of semistable objects undergo very well-behaved wall-crossing transformations.
2.1. Bridgeland stability conditions
We now consider more generally an admissible subcategory D\mathscr{D} of D^(b)(X)\mathrm{D}^{\mathrm{b}}(X) for a smooth and proper variety XX over a field kk, namely a full triangulated subcategory whose inclusion D↪D^(b)(X)\mathscr{D} \hookrightarrow \mathrm{D}^{\mathrm{b}}(X) admits both a left and a right adjoint. For instance, D=D^(b)(X)\mathscr{D}=\mathrm{D}^{\mathrm{b}}(X); otherwise we think of D\mathscr{D} as a smooth and proper noncommutative variety.
We fix a finite rank free abelian group Lambda\Lambda and a group homomorphism
from the Grothendieck group K_(0)(D)K_{0}(\mathscr{D}) of D\mathscr{D} to Lambda\Lambda.
Definition 2.1. A Bridgeland stability condition on D\mathscr{D} with respect to (v,Lambda)(v, \Lambda) is a pair sigma=(Z,P)\sigma=(Z, \mathcal{P}) where
Z:Lambda rarrCZ: \Lambda \rightarrow \mathbb{C} is a group homomorphism, called central charge, and
P=(P(phi))_(phi inR)\mathscr{P}=(\mathscr{P}(\phi))_{\phi \in \mathbb{R}} is a collection of full additive subcategories P(phi)subD\mathscr{P}(\phi) \subset \mathscr{D} satisfying the following conditions:
(1) for all nonzero E inP(phi)E \in \mathcal{P}(\phi), we have Z(v(E))inR_( > 0)*e^(i pi phi)Z(v(E)) \in \mathbb{R}_{>0} \cdot e^{i \pi \phi};
(2) for all phi inR\phi \in \mathbb{R}, we have P(phi+1)=P(phi)[1]\mathcal{P}(\phi+1)=\mathcal{P}(\phi)[1];
(3) if phi_(1) > phi_(2)\phi_{1}>\phi_{2} and E_(j)inP(phi_(j))E_{j} \in \mathcal{P}\left(\phi_{j}\right), then Hom(E_(1),E_(2))=0\operatorname{Hom}\left(E_{1}, E_{2}\right)=0;
(4) (Harder-Narasimhan filtrations) for all nonzero E inDE \in \mathscr{D}, there exist real numbers phi_(1) > phi_(2) > cdots > phi_(m)\phi_{1}>\phi_{2}>\cdots>\phi_{m} and a finite sequence of morphisms
such that the cone of s_(l)s_{l} is a non-zero object of P(phi_(l))\mathcal{P}\left(\phi_{l}\right);
(5) (support property) there exists a quadratic form QQ on Lambda_(R)=Lambda oxR\Lambda_{\mathbb{R}}=\Lambda \otimes \mathbb{R} such that
the kernel of ZZ is negative definite with respect to QQ, and
for all E inP(phi)E \in \mathcal{P}(\phi) for any phi\phi we have Q(v(E)) >= 0Q(v(E)) \geq 0;
(6) (openness of stability) the property of being in P(phi)\mathcal{P}(\phi) is open in families of objects in D\mathscr{D} over any scheme;
(7) (boundedness) objects in P(phi)\mathcal{P}(\phi) with fixed class v in Lambdav \in \Lambda are parameterized by a kk-scheme of finite type.
An object of the subcategory P(phi)\mathcal{P}(\phi) is called sigma\sigma-semistable of phase phi\phi, and sigma\sigma-stable if it admits no non-trivial subobject in P(phi)\mathcal{P}(\phi). The set of Bridgeland stability conditions on D\mathscr{D} is denoted by Stab(D)\operatorname{Stab}(\mathscr{D}), where we omit the dependence on (v,Lambda)(v, \Lambda) from the notation.
Conditions (1)-(4) form the original definition in [20] and ensure we have a notion of slope-stability. The support property is necessary to show that stability conditions vary continuously (see Theorem 2.2 below) and admit a well-behaved wall and chamber structure: fundamentally, this is due to the simple linear algebra consequence that given C > 0C>0, there are only finitely many classes w in Lambdaw \in \Lambda of semistable objects with |Z(w)| < C|Z(w)|<C. Conditions (6) and (7) were introduced in [73,74], with similar versions appearing previously in [42, sECTION 3]; they guarantee the existence of moduli spaces of semistable objects.
Theorem 2.2 (Bridgeland deformation theorem). The set Stab(D)\operatorname{Stab}(\mathscr{D}) has the structure of a complex manifold such that the natural map
Z:Stab(D)rarr Hom(Lambda,C),quad(Z,P)|->Z\mathcal{Z}: \operatorname{Stab}(\mathscr{D}) \rightarrow \operatorname{Hom}(\Lambda, \mathbb{C}), \quad(Z, \mathcal{P}) \mapsto Z
is a local isomorphism at every point.
For conditions (1)-(5), this is a reformulation of Bridgeland's main result [20, THEOREM 1.2]. It says that sigma=(Z,P)\sigma=(Z, \mathcal{P}) can be deformed uniquely given a small deformation of Zâ‡Z^(')Z \leadsto Z^{\prime}, roughly as long as Z^(')(E)!=0Z^{\prime}(E) \neq 0 remains true for all sigma\sigma-semistable objects EE. (More precisely, any path where QQ remains negative definite on Ker Z^(')\operatorname{Ker} Z^{\prime} can be lifted uniquely to a path in Stab(D)\operatorname{Stab}(\mathscr{D}).) With the additional conditions (6) and (7), Theorem 2.2 was proved in [73, THEOREM 3.20] and [67, SECTION 4.4], where the most difficult aspect is to show that openness of stability is preserved under deformations.
The theory has been developed over an arbitrary base scheme in [8]. A stability condition over a base is the datum of a stability condition on each fiber, such that families of objects over the base have locally constant central charges, satisfy openness of stability, and a global notion of HN filtration after base change to a one-dimensional scheme; moreover, we impose a global version of the support property and of boundedness. An analogue of Theorem 2.2 holds; differently to the absolute case, assuming the support property is not enough and the proof requires the additional conditions (6) and (7).
The construction of Bridgeland stability conditions is discussed in Section 4; in particular, they exist on surfaces and certain threefolds.
2.2. Stability conditions as polarizations
It was first suggested in the arXiv version of [21] to think of sigma\sigma as a polarization of the noncommutative variety D\mathscr{D}. We now review some results partly justifying this analogy. A polarization of a variety XX by an ample line bundle HH gives projective moduli spaces of HH-Gieseker-stable sheaves; the following two results provide an analogue.
Theorem 2.3 (Toda, Alper, Halpern-Leistner, Heinloth). Given sigma in Stab(D)\sigma \in \operatorname{Stab}(\mathscr{D}) and v in Lambdav \in \Lambda, there is a finite type Artin stack M_(sigma)(v)\mathcal{M}_{\sigma}(v) of sigma\sigma-semistable objects EE with v(E)=vv(E)=v and fixed phase. In characteristic zero, it has a proper good moduli space M_(sigma)(v)M_{\sigma}(v) in the sense of Alper.
Proof. The existence as Artin stack is [73, THEOREM 3.20], while the existence of a good moduli space is proven in [2, THEOREM 7.25]; See alSO [8, THEOREM 21.24].
Theorem 2.4. The algebraic space M_(sigma)(v)M_{\sigma}(v) admits a Cartier divisor â„“_(sigma)\ell_{\sigma} that has strictly positive degree on every curve. In characteristic zero, if M_(sigma)(v)M_{\sigma}(v) is smooth, or more generally if it has Q\mathbb{Q}-factorial log-terminal singularities, then M_(sigma)(v)M_{\sigma}(v) is projective.
Proof. The existence of the Cartier divisor and its properties is the Positivity Lemma in [12], see also [8, tHEOREM 21.25]. The projectivity follows from [76, COROLLARY 3.4].
As studied extensively in Donaldson theory in the 1990s, the Gieseker-moduli spaces may change as HH crosses walls in the ample cone.
Theorem 2.5. Fix a vector v in Lambdav \in \Lambda. Then there exists a locally finite union W_(v)\mathcal{W}_{v} of realcodimension one submanifolds in Stab(D)\operatorname{Stab}(\mathscr{D}), called walls, such that on every connected component C\mathcal{C} of the complement Stab(D)\\W_(v)\operatorname{Stab}(\mathscr{D}) \backslash \mathcal{W}_{v}, called a chamber, the moduli space M_(sigma)(v)M_{\sigma}(v) is independent of the choice sigma inâ„“\sigma \in \mathcal{\ell}.
Theorem 2.5 follows from the results in [21, SECTION 9]; see also [73, PROPOSITION 2.8] and [10, PROPOSITION 3.3]. The set W_(v)\mathcal{W}_{v} consists of stability conditions for which there exists an exact triangle A rarr E rarr BA \rightarrow E \rightarrow B of semistable objects of the same phase with v(E)=vv(E)=v, but v(A)v(A) not proportional to vv. Locally, the wall is defined by Z(A)Z(A) being proportional to Z(E)Z(E), and the objects EE is unstable on the side where arg(Z(A)) > arg(Z(E))\arg (Z(A))>\arg (Z(E)); often it is stable near the wall on the other side, e.g., when AA and BB are stable and the extension is nontrivial. The support property (5) is again crucial in the proof of Theorem 2.5: it constrains the classes a=v(A),b=v(B)a=v(A), b=v(B) involved in a wall, and locally that produces a finite set.
Sometimes, one can describe W_(v)\mathcal{W}_{v} completely, namely when we know which of the moduli spaces M_(sigma)(a)M_{\sigma}(a) and M_(sigma)(b)M_{\sigma}(b) are nonempty.
2.3. K3 categories
Such descriptions of W_(v)\mathcal{W}_{v} have been particularly powerful in the case of K3\mathrm{K} 3 categories; it has also been carried out completely for D^(b)(P^(2))\mathrm{D}^{\mathrm{b}}\left(\mathbb{P}^{2}\right), where the answer is more involved [24,54]. For this section, we work over the complex numbers and let D\mathscr{D} be
(1) the derived category D=D^(b)(S)\mathscr{D}=\mathrm{D}^{\mathrm{b}}(S) of a smooth projective K3\mathrm{K} 3 surface, or
of the derived category of a smooth cubic fourfold YY, or
(3) the Kuznetsov component of a Gushel-Mukai fourfold defined in [47].
In (1) we can also allow a Brauer twist; one expects further examples of Kuznetsov components of Fano varieties where similar results hold. In all these cases, D\mathscr{D} is a Calabi-Yau-2 category: there is a functorial isomorphism Hom(E,F)=Hom(F,E[2])^(vv)\operatorname{Hom}(E, F)=\operatorname{Hom}(F, E[2])^{\vee} for all E,F inDE, F \in \mathscr{D}.
Moreover, it has an associated integral weight two Hodge structure H^(**)(D,Z)H^{*}(\mathscr{D}, \mathbb{Z}) with an even pairing (_,_)\left(\_, \_\right); in the case of a K3\mathrm{K} 3 surface, H^(**)(D^(b)(S),Z)=H^(**)(S,Z)H^{*}\left(\mathrm{D}^{\mathrm{b}}(S), \mathbb{Z}\right)=H^{*}(S, \mathbb{Z}) with H^(0)H^{0} and H^(4)H^{4} considered to be (1,1)(1,1)-classes; in the other cases, the underlying lattice is the same, and after the initial indirect construction in [1] there is now an intrinsic construction based on the topological K-theory and Hochschild homology of D\mathscr{D} [64]. There is a Mukai vector v:K_(0)(D)rarrH^(1,1)(D,Z)v: K_{0}(\mathscr{D}) \rightarrow H^{1,1}(\mathscr{D}, \mathbb{Z}) satisfying chi(E,F)=-(v(E),v(F))\chi(E, F)=-(v(E), v(F)) for all E,F inDE, F \in \mathscr{D}.
In all three cases, there is a main component Stab^(†)(D)sub Stab(D)\operatorname{Stab}^{\dagger}(\mathscr{D}) \subset \operatorname{Stab}(\mathscr{D}) with an effective version of Theorem 2.2 for Lambda=H^(1,1)(D,Z)\Lambda=H^{1,1}(\mathscr{D}, \mathbb{Z}) : the map Z\mathbb{Z} is a covering of an explicitly described open subset of Hom(Lambda,C\operatorname{Hom}(\Lambda, \mathbb{C} ), see [21] for case (1), [8,9] for case (2), and [65] for case (3).
Now consider a family of such K3\mathrm{K} 3 categories, given by a family of K3\mathrm{K} 3 surfaces or Fano fourfolds over a base scheme, respectively. In this case, Mukai's classical deformation argument applies: every stable object EE in a given fiber is simple, i.e., it satisfies Hom(E,E)=C\operatorname{Hom}(E, E)=\mathbb{C}, and so Ext^(2)(E,E)=C\operatorname{Ext}^{2}(E, E)=\mathbb{C} by Serre duality; therefore the obvious obstruction to extending EE across the family, namely that v(E)v(E) remains a Hodge class, is the only one. Extending such deformation arguments to D\mathscr{D} was the original motivation for introducing stability conditions for families of noncommutative varieties, see [8, SECTION 31]. They allows us to deduce nonemptiness of moduli spaces from the previously known case of K3\mathrm{K} 3 surfaces (and simplify the previous classical argument for Gieseker stability on K3 surfaces by reduction to elliptically fibered K3\mathrm{K} 3 s, see [18]), which leads to the following result.
Theorem 2.6 (Mukai, Huybrechts, O'Grady, Yoshioka, Toda [8,12,65][8,12,65] ). Let v inH^(1,1)(D,Z)v \in H^{1,1}(\mathscr{D}, \mathbb{Z}) be primitive, and sigma inStab^(†)(D)\sigma \in \operatorname{Stab}^{\dagger}(\mathcal{D}) be generic. Then M_(sigma)(v)M_{\sigma}(v) is nonempty if and only if v^(2):=(v,v) >= -2v^{2}:=(v, v) \geq-2; in this case, it is a smooth projective irreducible holomorphic symplectic (IHS) variety.
More precisely, M_(sigma)(v)M_{\sigma}(v) is of K^([n])\mathrm{K}^{[n]}-type, where n=(v^(2)+2)//2n=\left(v^{2}+2\right) / 2, i.e., it is deformation equivalent to the Hilbert scheme of nn points on a K3 surface (see [25,34] for the basic theory of irreducible holomorphic symplectic varieties). If v^(2) >= 2v^{2} \geq 2, the Mukai morphism
induced by a (quasi)universal family gives an identification of H^(2)(M_(sigma)(v),Z)H^{2}\left(M_{\sigma}(v), \mathbb{Z}\right) with v^(_|_)v^{\perp}. If v^(2)=0v^{2}=0, then M_(sigma)(v)M_{\sigma}(v) is a K3\mathrm{K} 3 surface and H^(2)(M_(sigma)(v),Z)H^{2}\left(M_{\sigma}(v), \mathbb{Z}\right) is identified with v^(_|_)//vv^{\perp} / v.
Knowing exactly which semistable objects exist then allows us to describe exactly when we are on a wall. While a complete result as in [11, THEOREM 5.7] also needs to treat essential aspects of the wall-crossing behavior, the basic result is simple to state:
Theorem 2.7 ([11]). Let v inH^(1,1)(D,Z)v \in H^{1,1}(\mathscr{D}, \mathbb{Z}) be a primitive class. Then sigma=(Z,P)inStab^(†)(D)\sigma=(Z, \mathcal{P}) \in \operatorname{Stab}^{\dagger}(\mathscr{D}) lies on a wall for vv if and only if there exists classes a,b inH^(1,1)(D,Z)a, b \in H^{1,1}(\mathscr{D}, \mathbb{Z}) with v=a+bv=a+b, a^(2),b^(2) >= -2a^{2}, b^{2} \geq-2 and Z(a),Z(b)Z(a), Z(b) are positive real multiples of Z(v)Z(v).
And the fundamental reason is similarly simple to explain: by Theorem 2.6, this allows for the existence of extensions
{:(2.1)0rarr A rarr E rarr B rarr0:}\begin{equation*}
0 \rightarrow A \rightarrow E \rightarrow B \rightarrow 0 \tag{2.1}
\end{equation*}
where v(A)=a,v(E)=v,v(B)=bv(A)=a, v(E)=v, v(B)=b, and A,E,BA, E, B are all semistable of the same phase For stronger results, we need to know when such EE can become stable near the wall.
3. CONSTRUCTIONS BASED ON K3 CATEGORIES
In this section we present three applications of stability conditions on K3\mathrm{K} 3 categories, to irreducible holomorphic symplectic varieties, to curves, and to cubic fourfolds.
3.1. Curves in irreducible holomorphic symplectic manifolds
Let MM be a smooth projective irreducible holomorphic symplectic (IHS) variety of K3^([n])\mathrm{K3}^{[n]}-type, with n >= 2n \geq 2. We let q_(M)q_{M} be the Beauville-Bogomolov-Fujiki quadratic form on H^(2)(M,Z)H^{2}(M, \mathbb{Z}). By [25, sEction 3.7.1], there exists a canonical extension
of lattices and weight-2 Hodge structures, where the lattice widetilde(Lambda)_(M)\widetilde{\Lambda}_{M} is isometric to the extended K3\mathrm{K} 3 lattice U^(o+4)o+E_(8)(-1)^(o+2)U^{\oplus 4} \oplus E_{8}(-1)^{\oplus 2}. Let us denote by v in widetilde(Lambda)_(M)v \in \widetilde{\Lambda}_{M} a generator of vartheta(H^(2)(M,Z))^(_|_)\vartheta\left(H^{2}(M, \mathbb{Z})\right)^{\perp} : it is of type (1,1)(1,1) and square v^(2)=2n-2v^{2}=2 n-2. The lattice widetilde(Lambda)_(M)\widetilde{\Lambda}_{M} is called the Markman-Mukai lattice associated to MM. If M=M_(sigma)(v)M=M_{\sigma}(v), for a stability condition sigma inStab^(†)(D^(b)(S))\sigma \in \operatorname{Stab}^{\dagger}\left(\mathrm{D}^{\mathrm{b}}(S)\right) on a K3\mathrm{K} 3 surface SS, then widetilde(Lambda)_(M)=H^(**)(S,Z)\widetilde{\Lambda}_{M}=H^{*}(S, \mathbb{Z}) with the Mukai pairing, the notation for the vector vv is coherent, and vartheta_(M)\vartheta_{M} is the Mukai morphism mentioned after Theorem 2.6.
We let Pos(M)\operatorname{Pos}(M) be the connected component of the positive cone of MM containing an ample divisor class:
The following result rephrases and proves a conjecture by Hassett-Tschinkel and gives a complete description of the ample cone of MM.
Theorem 3.1. Let MM be a smooth projective IHS variety of K^([n])\mathrm{K}^{[n]}-type. The ample cone of MM is a connected component of
Theorem 3.1 is proved in [11] for moduli spaces of stable sheaves/complexes on a K3\mathrm{K} 3 surface, and extended in [7] to all IHS of K3^([n])\mathrm{K} 3{ }^{[n]}-type, by using deformation theory of rational curves on IHS varieties.
The approach to Theorem 3.1 via wall-crossing is as follows. Let SS be a K3 surface and M=M_(sigma_(0))(v)M=M_{\sigma_{0}}(v) be a moduli space of sigma_(0)\sigma_{0}-stable objects in D^(b)(S)\mathrm{D}^{\mathrm{b}}(S), where v inv \inH^(1,1)(D^(b)(S),Z)H^{1,1}\left(\mathrm{D}^{\mathrm{b}}(S), \mathbb{Z}\right) is a primitive vector of square v^(2) >= 2v^{2} \geq 2. As sigma\sigma varies in the chamber varphi\varphi containing sigma_(0)\sigma_{0}, Theorem 2.4 gives a family of ample divisor classes â„“_(sigma)\ell_{\sigma} in Pos(M)\operatorname{Pos}(M). When sigma\sigma reaches a wall of â„“\mathcal{\ell}, as given by Theorem 2.7, the class â„“_(sigma)\ell_{\sigma} remains nef. On the other hand, consider an object EE that becomes strictly semistable on the wall, admitting an exact
sequence as in (2.1). Varying the extension class in a line in P(Ext^(1)(B,A))\mathbb{P}\left(\operatorname{Ext}^{1}(B, A)\right) produces a P^(1)\mathbb{P}^{1} of such objects, and Theorem 2.4 shows that â„“_(sigma)\ell_{\sigma} has degree zero on this curve. We have found an extremal curve and, dually, a boundary wall of the ample cone.
FIGURE 1
The approach to Theorem 3.1.
We summarize the history underlying Theorem 3.1 with the diagram in Figure 1. The analogue of Theorem 2.6 for Gieseker-stable sheaves involves a two-step argument, using autoequivalences and deformations. Wall-crossing techniques then imply the existence of Bridgeland stable objects on K3 surfaces, and thus Theorem 2.6. As discussed above, a finer wall-crossing analysis based on Theorem 2.7 then produces the extremal rational curves on moduli spaces that appear implicitly as extremal curves in Theorem 3.1. Finally, another deformation argument, involving rational curves, deduces Theorem 3.1 for all IHS manifolds of K3^([n])\mathrm{K} 3{ }^{[n]}-type. Wall-crossing combined with stability conditions in families can also simplify the approach to Theorem 2.6, see [18].
3.2. Curves
Consider a Brill-Noether (BN) wall in Stab(D^(b)(X))\operatorname{Stab}\left(\mathrm{D}^{\mathrm{b}}(X)\right) for a variety XX : the structure sheaf O_(X)\mathcal{O}_{X} is stable and of the same phase phi\phi as objects EE of a fixed class vv. Then O_(X)\mathcal{O}_{X} is an object of the abelian category P(phi)\mathcal{P}(\phi) with no subobjects; hence the evaluation map O_(X)ox\mathcal{O}_{X} \otimesH^(0)(E)rarr EH^{0}(E) \rightarrow E must be injective, giving a short exact sequence
{:(3.1)0rarrO_(X)oxH^(0)(E)rarr E rarr Q rarr0:}\begin{equation*}
0 \rightarrow \mathcal{O}_{X} \otimes H^{0}(E) \rightarrow E \rightarrow Q \rightarrow 0 \tag{3.1}
\end{equation*}
where Q inP(phi)Q \in \mathscr{P}(\phi) is also semistable. Applying known inequalities for Chern classes of semistable objects to ch(Q)=v-h^(0)(E)ch(O_(X))\operatorname{ch}(Q)=v-h^{0}(E) \operatorname{ch}\left(\mathcal{O}_{X}\right) can directly lead to bounds on h^(0)(E)h^{0}(E).
This simple idea turns out to be powerful. For a K3\mathrm{K} 3 surface SS, we can be more precise: applying Theorem 2.6 to the class of QQ, we can construct all EE with given r=h^(0)r=h^{0} as a Grassmannian bundle Gr(r,Ext^(1)(Q,O_(S)))\operatorname{Gr}\left(r, \operatorname{Ext}^{1}\left(Q, \mathcal{O}_{S}\right)\right) of extensions over the moduli space of such QQ.
Corollary 3.2. Let SS be a K3K 3 surface, v inH^(1,1)(S,Z)v \in H^{1,1}(S, \mathbb{Z}) primitive and sigma\sigma be a stability condition near the Brill-Noether wall for vv. If the lattice generated by vv and v(O_(S))v\left(\mathcal{O}_{S}\right) is saturated, then the locus of objects E inM_(sigma)(v)E \in M_{\sigma}(v) with h^(0)(E)=rh^{0}(E)=r has expected dimension.
In [5], this is applied, in the case where Pic(S)=Z*H\operatorname{Pic}(S)=\mathbb{Z} \cdot H, to rank zero classes of the form v=(0,H,s)v=(0, H, s). In this case, there are no walls between the BN\mathrm{BN} wall and the large volume limit; hence Corollary 3.2 applies in the large volume limit, and thus to zero-dimensional torsion sheaves supported on curves in the primitive linear system. This gives a variant of Lazarsfeld's proof [49] of the Brill-Noether theorem: every curve in the primitive linear system is Brill-Noether general.
This approach has been significantly strengthened in [27]: instead of requiring EE to be semistable near the Brill-Noether wall, it is sufficient to control the classes occurring in its HN filtration. A bound on h^(0)h^{0} is obtained by applying Corollary 3.2 to all HN filtration factors. Thus we need to consider a point near the Brill-Noether walls for all HN\mathrm{HN} factors, and which is the limit point where Z(O_(X))∼0Z\left(\mathcal{O}_{X}\right) \sim 0.
Proposition 3.3 ([27, PRoposition 3.4]). Let SS be a K3 surface of Picard rank one. There exists a limit point bar(sigma)\bar{\sigma} of the space of stability conditions, with central charge bar(Z)\bar{Z}, and a constant CC, such that for (most) objects in the heart, we have
h^(0)(E)+h^(1)(E) <= C*sum_(l)|( bar(Z))(E_(l)//E_(l-1))|h^{0}(E)+h^{1}(E) \leq C \cdot \sum_{l}\left|\bar{Z}\left(E_{l} / E_{l-1}\right)\right|
where E_(0)subE_(1)sub cdots subE_(m)E_{0} \subset E_{1} \subset \cdots \subset E_{m} is the HNH N filtration of EE near bar(sigma)\bar{\sigma}.
The following application completes a program originally proposed by Mukai [62]:
Theorem 3.4 ([27, 28]). Let SS be a polarised K3K 3 surface with Pic(S)=Z*H\operatorname{Pic}(S)=\mathbb{Z} \cdot H and genus g >= 11g \geq 11, and let C in|H|C \in|H|. Then SS is the unique K3 surface containing CC, and can be reconstructed as a Fourier-Mukai partner of a Brill-Noether locus of stable vector bundles on CC with prescribed number of sections.
The structure of the argument is as follows. The numerics are chosen such that there is a two-dimensional moduli space hat(S)\hat{S}, necessarily a K3 surface, of stable bundles EE on SS whose restriction E|_(C)\left.E\right|_{C} is automatically in the Brill-Noether locus. Conversely, given a stable bundle VV on CC, its push-forward i_(**)Vi_{*} V along i:C↪Si: C \hookrightarrow S is stable at the large volume limit. Standard wall-crossing arguments bound its HN filtration near the limit point bar(sigma)\bar{\sigma} in Proposition 3.3, which then gives a bound on h^(0)(V)h^{0}(V). The argument also shows that equality-the Brill-Noether condition-only holds for the HN\mathrm{HN} filtration E rarr E|_(C)=i_(**)V rarr E(-H)[1]\left.E \rightarrow E\right|_{C}=i_{*} V \rightarrow E(-H)[1], i.e., when VV is the restriction of a vector bundle in hat(S)\hat{S}. Thus hat(S)\hat{S} is a Brill-Noether locus on CC, and SS can be reconstructed as a Fourier-Mukai partner of hat(S)\hat{S}.
3.3. Surfaces in cubic fourfolds
Let Y subP^(5)Y \subset \mathbb{P}^{5} be a complex smooth cubic fourfold and let hh be the class of a hyperplane section. Following [38], we say that YY is special of discriminant dd, and write Y inC_(d)Y \in \mathscr{C}_{d}, if there exists a surface Sigma sub Y\Sigma \subset Y such that h^(2)h^{2} and Sigma\Sigma span a saturated rank two lattice in H^(4)(Y,Z)H^{4}(Y, \mathbb{Z}) with
